Aha, your comment just put into focus for me what was unclear in the original post. I wasn't claiming that the Romans didn't have a formal system for analyzing their own grammar as well as systems for outsiders to learn it, presumably involving a lot of memorization. But there's a fundamental shift between the ways that people analyze a living language and the way they analyze a dead one--it stops being about "how are people using the language now" and becomes "how were people using the language then" and a lot of the teaching pedagogy for Latin feels like it is significantly better suited for learning a dead one.

I can imagine a parent explaining to their child the difference between the kinds of time words that get the ablative and the kinds of time words that get the accusative, etc. I can't imagine them demanding that their children learn whether a particular ablative is an ablative of origin or an ablative of accordance. They were metaphorically similar and you don't need to distinguish between them until there are no native speakers left and you need to compile a fetishistic listing of every way the language was used and force people to memorize it.

I feel like there's an interesting story in there, and that's what I want to poke at. So far this bibliography looks like it'll be a good place to start.