The biblical writings are pretty diverse and the Bible is much more like an anthology than a book, but in broad brush strokes, I’d say the writings are primarily narrative and narrative-adjacent. In other words, they are records of experience peppered with other things (songs, etc.).
I don’t think of very much of the biblical writings as being primarily theological in nature in the sense of abstracting systematic truth statements about God. That does happen, of course, here and there, and certainly there’s a lot of commentary on events that presupposes certain understandings about God, but for the most part the Scriptures don’t provide a systematic theology. That is something readers create.
Because the Scriptures are primarily a narrative, interpreters deal with that in various ways, and we see the first flash point in the early Gentile church - what does a predominantly Jewish narrative have to do with them? What do these Scriptures mean to Greeks and Egyptians who never had any connections to the Temple or the covenant at Sinai?
I think what Andrew was trying to do was map out in broad brush strokes the main categories of how early interpreters approached that issue and what the modern inheritance of those streams looks like. Maybe, as you say, the categories are too broad or it’s misleading to try to peg this or that person or group solely in one stream or the other. But I’m not convinced it’s nonsensical to talk about those categories as they relate to the history of interpretation. It’s no more nonsensical to talk about “allegorical interpretation” than it is to talk about a “doctrine of justification.” It’s a systematic abstraction of the historical data.
Ironically, you and Andrew may have some common ground, here, because a lot of what Andrew writes about is how this systematic abstraction obscures the actual historical angle on the Scriptures. Maybe you feel the same way about abstracting over the history of biblical interpretation.