You statically linked GTK+ won't work with the users UI styles though and probably won't support everyone's input methods or accessibility tools.

Static linking itself doesn't prevent modules. There's https://github.com/pikhq/musl-nscd for example

Not inherently, but static linking to glibc will not get you there without substantial additional effort, and static linking to a non-glibc C library will by default get you an absence of NSS.