Or maybe we should build an LSP that exposes an AST in a way that is more useful to IDEs, like IDEAs plugins do today.

In fact, the code in IDEA that does this is open source, so you can start with it, and build a better LSP based on it.

This would be an interesting idea, provided that editors go all-in on supporting such refactoring logic on the client side. The IDEA code defines "extension points" so language specific support for common actions can be implemented in a modular fashion. When the user decides to execute a refactoring, it is just the matter of iterating through the loaded extension points and choosing the one for the language being edited. Having worked with the IDEA code myself to build an LSP server, there end up being language-specific parts such that the support between e.g. Java and Kotlin have to be handled separately. However that was probably due to the way I wrote the server; these could be generalized to a single "refactor" entrypoint with some work. It's just that using the current LSP model with the server being the part with AST awareness, there isn't much gain in doing so.