For a real statically-linked linux system, the main issue is GPU support: you must relink all apps _really using_ a GPU, that to include the required GPU drivers.
With sound, alsa, it is fine since there is IPC/shared-memory based mixing that whatever the playback/capture devices [dmix/dsnoop]. Static linking is reasonable. (pulseaudio[012] IPC interfaces are bloaty kludges, hardly stable in time, 0..1..2.., not to be trusted compared to the hardcore stability of alsa one able to do a beyond good enough job *and* _real_ in-process low latency hardware access at the same time).
x11 and wayland are IPC based, then no issue here neither.
But for the GPU, we would need a wayland vulkan3D-inspired set of IPC/shared-memory interfaces (with a 3D enabled wayland compositor). For compute, the interfaces would be de-coupled from the wayland compositor (shared dma-buffers).
The good part of this would be to free our system interfaces from the ultra complex ELF (one could choose an excrutiatingly simple executable file format, aka a modern executable file format, but will need compilers/linkers support to help legacy support).
There is a middle ground though: everything statically linked, except the apps requiring the GPU driver (for that ELF is grotesquely overkill), still provided as a shared library.