Found out today that apparently Godot is in fact capable of exporting zips on build. Previously, I was zipping the builds manually using a script. However, if you just change export_path to a .zip file, then Godot will zip the build for you.

For example, I previously had the path ../bin/linux-no-signal.x86_64 for my Linux build. Changing this to ../bin/linux-no-signal.zip zips the executable for me. In fact, on Windows, it will zip the executable with the executable bit set which would otherwise be lost if I exported the .x86_64 file and then zipped it manually later.