Can you build GUI programs with this? I'm thinking anything that would depend on GPU drivers. Anything built with SDL, OpenGL, Vulkan, whatever.

No, in my experimentation I tried to convert OBS into static and it had the issue of it's gui not working. I am not exactly sure what's the reason but maybe you can check out another library like sdl etc. that you mention, I haven't tested out SDL,OpenGL etc's support to be honest but I think that maybe it might not work in the current stage or not (not sure), there is definitely a possibility of making it possible tho because CLI applications work just fine (IO and everything) so I am not really sure what caused my obs studio error but perhaps you can try it and then let me know if you need any help/share the results!