Karl Barth's excellent explanation of three temptations of Jesus by Satan in the wilderness in the small print of his extended discussion of "§59.2 The Judge Judged in Our Place" of his Church Dogmatics IV.1, may be the best exegesis of this event I've encountered. (I've included a long quote of the text below.)

1) In succumbing to the first temptation, Barth explains, that Jesus would have failed to be truly man and for us. He would demonstrate that man could not be saved, if Jesus himself could not endure as a man. Failure meant that he was unable to live out the righteous life demanded of us because he could not endure temptation for even himself. How may man be saved, if even Jesus could not remain a sinless man without divine intervention? He would not be able to restore true humanity because he would have failed to live with our very fallen nature, so that would mean that no one would be able to live out the righteous life, and there would be no hope. If Jesus was unable to endure his own temptations as a Man, then he certainly could not have done so for others. 

2) In succumbing to the second temptation, Jesus would not have overcome the world by becoming its leader, but would have only empowered an evil kingdom to endure eternally by becoming its highest leader. By ruling all the kingdoms, Jesus would establish these evil powers permanently and universally, and in opposition to the Kingdom of God. Jesus would not have been able to establish a new kingdom that would overthrow evil dominions that existed by ruling within that evil kingdom. In here, there are statements of the confession church who lived under the shadow of Nazi Germany.

3) Barth finds the final temptation the most mesmerizing of the three with analogies to temptation of Jesus in the Garden of Gethsemane, and also to Kierkegaard's leap of faith into the unknown. If Jesus succumbs to the temptation to jump from the temple heights, then he does so only to demonstrate to himself and to Satan that he is the Son of God and the Messiah, and reveals that he does not believe that he is who he has believed to be, and if he is unable to know that he is the divine son, then how can fulfill his passion for others? This temptation and the passion of Jesus would become a selfish act, to self-vindicate Jesus as the Son of God, rather than a Messianic act of deliverance, victory and atonement for us. Barth wisely notices that in the Passion, Jesus essentially does what Satan has tempted him to do... to leap into the unknown in the death of his Passion.

By succumbing in any of these three temptations, Jesus would cease to be Emmanuel, no longer God with us, and no longer God for us.

In both Evangelists the first Satanic suggestion is that after the forty days of hunger He should change the stones of the wilderness into bread in the power of His divine Sonship by His Word. What would it have meant if Jesus had yielded? He would have used the power of God which He undoubtedly had like a technical instrument placed at His disposal to save and maintain His own life. He would then have stepped out of the series of sinners in which He placed Himself in His baptism in Jordan. Of His own will He would have abandoned the role of the One who fasts and repents for sinners. He would have broken off His fasting and repentance in the fullness of divine power and with the help of God, but without consulting the will and commandment of God, because in the last resort His primary will was to live. He would have refused to give Himself unreservedly to be the one great sinner who allows that God is in the right. to set His hopes for the redemption and maintenance of His life only on the Word of God. in the establishment of which He was engaged in this self-offering. He would have refused to be willing to live only by this Word and promise of God, and therefore to continue to hunger. In so doing He would, of course, only have done what in His place and with His powers all other men would certainly have done. From the standpoint of all other men He would only have acted reasonably and rightly. "Rabbi, eat" is what His disciples later said to Him (John 4:31) quite reasonably and in all innocence. But then He would not have made it His meat "to do the will of him that sent him, and to finish his work" (John 4:34). Instead of acting for all other men and in their place. He would have left them in the lurch at the very moment when He had made their cause His own. Jesus withstood this temptation. He persisted in obedience. in penitence, in fasting. He hungered in confidence in the promise of manna with which the same God had once fed the fathers in the wilderness after He had allowed them to hunger (Deut 8:3). He willed to live only by that which the Word of God creates, and therefore as one of the sinners who have no hope apart from God, as the Head and King of this people. His decision was, therefore, a different one from that which all other men would have taken in His place, and in that way it was the righteousness which He achieved in their stead.

According to Luke, the second Satanic suggestion is that Satan, to whom the world belongs, should give him lordship over it, at the price of His falling down and worshiping him. What would it have meant if Jesus had done this? Obviously He would have shown that He repented having received the baptism of John and that He did not intend to complete the penitence which He had begun. He would have ceased to recognize and confess the sin of the world as sin, to take it upon Himself as such, and in His own person to bring to an issue the conflict with it (as with man's contradiction against God and himself). He would have won through and been converted to a simpler and more practical and more realistic approach and way. He would have determined to drop the question of the overcoming and removing of evil, to accept the undeniable fact of the overlordship of evil in the world, and to do good, even the best, on this indisputable presupposition, on the ground and in the sphere of this overlordship. Why not set up a real kingdom of God on earth? an international order modelled on the insights of Christian humanitarianism, in which, of course, a liberal-orthodox, ecumenical, confessional Church might also find an appropriate place? Note that to do this He was not asked to renounce God or to go over to atheism. He had only to lift His hat to the usurper. He had only to bow the knee discreetly and privately to the devil. He had only to make the quiet but solid and irreversible acknowledgment that in that world of splendour the devil should have the first and final word, that at bottom everything should remain as it had been. On this condition we can all succeed in the world, and Jesus most of all. In the divine and human kingdom set up on this condition there would have been no place for the cross. Or rather, in this world ostensibly ruled by Jesus but secretly by Satan, the cross would have been harmlessly turned into a line and profound symbol: an ornament in the official philosophy and outlook; but also an adornment (e.g., an episcopal adornment) in the more usual sense of the word; a suitable recollection of that which Jesus avoided and which is not therefore necessary for anyone else. What other man in Jesus' place would not have been clever enough to close with this offer? But what He had to do and willed to do in place of all would not then have been done. He would again have left them in the lurch and betrayed them, in spite of all the fine and good things that the world-kingdom of Satan and Jesus might have meant for them. For of what advantage is even the greatest glory to a world which is still definitively unreconciled with God? Of what gain to man are all the conceivable advantages and advances of such a kingdom? But Jesus resisted this temptation too. He refused to be won over to this attractive realism. As the one great sinner in the name and place of all others, without any prospect of this glory, quite unsuccessfully, indeed with the certainty of failure, He willed to continue worshipping and Serving God alone. He willed to persist in repentance and obedience. This was the righteousness which He achieved for us.

The third temptation, according to Luke’s account, is the most astonishing of all. The dignity of the setting, die temple of God in the holy city of Jerusalem, is obviously incomparably greater than the still secular dignity of that high mountain from which Jesus was shown and offered all the kingdoms of the world. It is of a piece that Satan now appears as an obviously pious man who can even quote the Psalms of David, and he gains in the seriousness and weight of his approach. Above all, his suggestion--we can hardly describe it by the horrible word temptation--is quite different from everything that has preceded it. It now consists in the demand to commit an act of supreme, unconditional, blind, absolute, total confidence in God--as was obviously supremely fitting for the Son of God. We might almost say. an act in the sense of and in line with the answers which Jesus Himself had given to the first two temptations, to live only by the Word of God, to serve and worship Him alone. In the last decades we have become accustomed to think of the seeking and attaining of totalitarian dominion as the worst of all evils, as that which is specifically demonic. But if the climax in Luke is right, there is something even worse and just as demonic. It is not just a matter of a miraculous display to reveal the Messiahship of Jesus. It is often interpreted in this way, but by a reading into the text rather than out of it. The text itself makes no mention whatever of spectators. It is rather a question of the testing and proving, of the final assuring of His relationship to God in foro conscientiae (in the sphere of his conscience), in the solitariness of man with God. Jesus is to risk this headlong plunge with the certainty, and to confirm the certainty, that God and His angels are with Him and will keep Him. Schlatter has rather mischievously said that what we have here is what is so glibly described "in contemporary theological literature" as the "leap" of faith. It certainly does seem to be something very like "existence in transcendence," or "the leap into the unknown," or in Reformation language ‘justification by faith alone,"justification in the sense that (in face of death and the last judgment. and in the hope that in trust in God these cart be overcome) than presumes to take it into his own hands, to carry it through as the work of his own robust faith, and in that way to have a part in it and to be certain of it; just as Empedocles (we do not know exactly why. but seriously and with courage) finally flung himself into the smoking crater of Etna, which is supposed to have thrown out again only his sandals; just as on this very same rock of the temple, when it was stormed by the Romans in A.D. 70, the last of the high priests put themselves to death with their own hands, possibly in despair, possibly in the hope that there would be a supreme miracle at that last hour. What would it have meant if Jesus had taken this leap? Note the remarkable closeness of the temptation to the way which Jesus did in fact tread. In this respect the Lucan order, in which this is the last and supreme temptation, is most edifying. He will "dare the leap into the abyss, the way to the cross, when the will of God leads Him to it" (Schlatter) . But what would have led Him to it here would have been His own will to make use of God in His own favour. He would have experimented with God for His own supreme pleasure and satisfaction instead of taking the purpose of God seriously and subjecting Himself to His good-pleasure and command. He would have tried triumphantly to maintain His lightness with God instead of persisting in penitence, instead of allowing God to be in the right against Him. In an act of supreme piety, in the work of a mystical enthusiasm. He would have betrayed the cause of God by making it His own cause, by using it to fulfill His own self-justification before God. If He had given way to this last and supreme temptation He would have committed the supreme sin of tempting God Himself, i.e., under the appearance of this most robust faith in Him demanding that He should accept this Jesus who believes so robustly instead of sinful man by Him and in His person. He would have demanded that He should be the most false of all false gods. the god of the religious man. And in so doing He would Himself have withdrawn from the society of sinful men as whose Representative and Head He was ordained to live and act. He would have left in the lurch the world unreconciled with God. "Farewell, O world, for I am weary of thee." But again we may ask. what other man, all things considered, would not actually have done this in His place? For Adamic man reaches his supreme form in religious self-sacrifice as the most perfect kind of self-glorification, in which God is in fact most completely impressed into the service of man, in which He is most completely denied under cover of the most complete acknowledgment of God and one’s fellows. Jesus did not do this. He rejected the supreme ecstasy and satisfaction of religion as the supreme form of sin. And in so doing He remained faithful to the baptism of John. He remained the One in whom God is well pleased. He remained sinless He remained in obedience. In our place He achieved the righteousness which had to be achieved in His person for the justification of us all and for the reconciliation of the world with God, the only righteousness that was necessary.

 We cannot ignore the negative form in which the righteousness of God appears in the event handed down in these passages. This is unavoidable, because we have to do with it in the wilderness, in the kingdom of demons, in the world unreconciled with God, and in conflict with that world. It is unavoidable because what we have here is a prefiguring of the passion. But in the passion, and in this prefiguring of it, the No of God is only the hard shell of the divine Yes, which in both cases is spoken in the righteous act of this one man. That this is the case is revealed at the conclusion of the accounts in Mark and Matthew by the mention of the angels who, when Satan had left Him, came and ministered unto Him. The great and glorious complement to this at the conclusion of the passion is the story of the resurrection.

Karl Barth, "Church Dogmatics IV.1, The Doctrine of Reconciliation, Section 59", pgs 261-264 [pgs 254-256 in study edition 21]