> If the program is statically linked, then nothing can happen

Static libs are embedded into most object-file formats (e.g. ELF) as isolated compilation units. Could you not just replace the static library within the object-file at the linkage level, in about the same way that old "resource editors" could replace individual resources within object-files, or the same way that programs like mkvmerge can replace individual streams within their target media-container format?

Static vs. dynamic linking is just about whether the top-level symbol table of the executable can be statically precomputed for that linkage. It doesn't impede you from re-computing the symbol table. That's what the linker already does, every time it links two compilation units together!