In the literature on Anselm, I’ve noticed what seems to me to be a real anxiety to make sure that Anselm isn’t “really” trying to get all the way to the necessity of the Incarnation by “pure reason.” The reason this explanation is necessary in the first place is that Anselm certainly appears to be doing that and doesn’t seem to view the attempt as problematic on a methodological level. In doing so, he is following in a proud tradition — for instance, Gregory of Nyssa’s “Great Catechism” is able to get to the Trinity and even to the creation and fall by means of something like the common sense of the Hellenistic world, though he recognizes that the Incarnation is going to be difficult to swallow. His general principle is to use Scripture for those who respect Scripture, and reason for those who accept only reason. Convincing the former is thought to be easier (though the historical record doesn’t seem to bear that out), but there isn’t a sharp division between the two that I can see.
I think that the reason for the anxiety about Anselm’s approach is that people are reading it in terms of Aquinas’s nature/grace distinction — i.e., reason can get you to a certain point (where Aristotle winds up), and then you need revelation, which is not contrary to reason but whose contents couldn’t be predicted using reason alone. The Trinity, for example, is firmly on the “revealed” side of this distinction, yet Gregory and Anselm both appear not to be worried about the fact that their reasoned argument has gone way over the line.
The reason for their lack of concern is probably that that line wasn’t a big concern of theirs, and we don’t need to read them anachronistically as though they knew about the nature/grace problem and were really concerned not to be doing something like “natural theology” because that would be somehow impious. Instead, maybe we should read them as doing what they’re actually doing — that is, assuming that the world described in Christian revelation is actually this very world where we are. If that is the case, then of course reason should be able to recognize the inner necessity of God’s actions in the world, because God is after all acting in this very same world where our reason finds its home.
I have a bunch of things that I want to say here but can’t fully support yet. For instance, I object to Aquinas’s two-tiered system first of all because it’s so inelegant. Another thing: maybe Barth’s polemic against “natural theology” should’ve gone further and also rejected the kind of “revealed theology” that’s defined in opposition to “natural theology.” Etc. I’m aware that there are all kinds of nuances that I’m not capturing here — sorry about that.