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 b…
To be fair ELF is complex mostly because of relocations, which are not purely to support shared libraries but also the nowadays ubiquitous PIE. But GPU drivers is a good point; I don't believe you can even statically link them today, you would only be statically linking a shim that tries to find the real driver at runtime.
About the shared libs, well, they are the utility shared libs, and the system interface shared libs. With a mostly statically linked elf/linux distro, all the utility libs would be statically linked, and the system interface shared libs would be statically linked if they have an IPC/shared-mem interface. In the end, only the GPU driver is an issue, namely would stay a shared libs.