If I'm not building a shared library, what is the benefit of having my project as a package? It seems Rye only supports python packages, but most my stuff never touches PyPI or anything similar.
Rye is a python only thing, and if you're using python without any external dependencies, it probably doesn't have that much benefit (although it does make having different python versions for different projects a lot cleaner if you need that).
I think that it has a huuuuge benefit for applications as well as libraries. Rye generates lock files, which are really handy for pinning down versions of something running in production so that it'll behave predictably.