> A lot of things have installed it to their own directory in appdata or program files

This is the windows way. In the early days (Windows 95/98) it was indeed common practice to put shared dlls and exes in C:\Windows or C:\Windows\System32, but without central oversight this was a nightmare: version conflicts were rampant, and uninstallers didn't know what could be safely removed. Everyone switched to installing everything into the program's install dir, and the world was much better off.

Some poweruser tooling gets fancy by checking if ffmpeg is installed systemwide, and if not downloads it to the programs appdata folder. But that exposes you to versioning issues if that version is too old or too new

This is what's always confused me about the Unix FHS. A unified hierarchy makes (un)install complex and error-prone indeed. One of my favorite things about Nix is that it threw FHS away.