I have had a lot of success with building against old glibc version and then bundling all required libraries except system ones (glibc, libGL, libX11 etc). Anything optional needs to be loaded with dlopen/dlsym but for many things that detail is already handled by libraries like SDL. That's generally what seems to be the common solution for games (well used to be at least, today most just build against the Steam runtime and call it a day). It's really not that different from Windows IMO except that the set of base system libs that you can rely on to be backwards compatible and don't need to bundle is smaller.

You can still pacakge that distro-agnostic build up into distro-specific package formats to make updates easier for users without needing zillion of different build roots just for Linux.