One way to put the question of the theory of the atonement might seem be to ask with Anselm: "Why did God become man?"
I do not think that that is the right way, though. For God is omnirational. This means that whenever he acts, he is moved by all the unexcluded good reasons for the action. But almost any good that is achieved by a morally permissible action will provide God with an unexcluded good reason for that action. Very high up among the goods achieved by God through the Incarnation is ensuring that we can be forgiven with justice. But there are many other goods achieved by the Incarnation: Aquinas lists nine other great goods. And besides the great goods, there are far lesser but nonetheless genuine goods, such as cooperating with Joseph in the making of tables. Plausibly, there are many goods in between the really great ones and the far lesser ones, such as providing Israel with a king, bringing the nations to Israel in worship, and healing certain lepers through a face-to-face interaction.
One might try to run some kind of a distinction between the goods wrought in the Incarnation, so as to delineate the goods that are the proper subject of a theory of the atonement or the proper answers to "Why did God become man?" Let's try a few such distinctions.
First, there is a distinction between those goods that could be achieved without God becoming man and those goods that can only be achieved by God becoming man. But this won't draw the distinguishing line in the right place. First, pace Anselm, it is far from clear that we couldn't be forgiven with justice simply by divine fiat. Certainly, Aquinas thought we could. It would be fitting for the penalty to be paid by Christ on our behalf, but it would not be unjust for God simply to release us. Second, some of the goods lower down on the list require the Incarnation. Thus, the good of God's cooperating with Joseph in carpentry (this cooperation makes me think of Enoch walking with God, but in the case of Joseph there is a greater literalness) requires an Incarnation. (God can, by his omnipotent power, create tables ex nihilo, but such creation is not carpentry.) Likewise, there is a value to God healing the lepers through a face-to-face interaction of a sort that can only be had by means of the Incarnation.
Second, perhaps we could distinguish between reasons that are such that if they were present on their own, God would still have decided on the Incarnation, and reasons that were only contributory. Thus, while omnirationality implies that God became man in part in order to work with Joseph, surely if no other goods were realized by the Incarnation than God's engaging in carpentry with Joseph, then there would be no Incarnation. But why think that? God could have done it. Maybe it would be an unlikely scenario, but surely a possible one. Why accept the counterfactual of free will that had God had working with Joseph as the only reason to be Incarnate then he wouldn't have done it? Such a counterfactual seems meaningless (or trivially false) to me, and even Molinists need not extend their view to counterfactuals of divine freedom.
Third, perhaps we can distinguish reasons such that were they absent, the Incarnation would not have occurred. But again we get into dubious counterfactuals about divine decisions. Take one of the "big" reasons for the Incarnation. I see no reason to accept the counterfactual that had that reason been absent, there would have been no Incarnation. God might still have become incarnate for the other reasons.
There is, of course, a distinction as to the weight of the reasons. The "big" reasons are much better reasons. But this distinction is one of degree.
So it seems to me that the answer to the question of why God became man is simply a list. Aquinas gives ten items to put on the list. There are many more. We can prioritize the items on the list, of course. But each of the items on the list will be a reason that God was acting in the light of, since God acts in the light of all the unexcluded good reasons in favor of his action.
This does not imply that there is no such thing as a theory of the atonement, just that the question "Why did God become man?" doesn't delineate the theory precisely enough.