Refactor method dispatch infrastructure

This is a pretty major refactoring of the method dispatch infrastructure. It is intended to avoid gross inefficiencies and enable caching and other optimizations (e.g. #17995), though it itself doesn't seem to execute particularly faster yet. It also solves some cases where we were failing to resolve methods that we theoretically should have succeeded with.

Fixes #18674.

cc #18208 


If we're no longer using tys, why keep it around?