Steam doesn’t grab binaries from $PATH, it uses .desktop files to figure out what can be executed, which live in the various directories under $XDG_DATA_DIRS. You’d need to pass that variable to steam somehow, no idea how it processes it or if there are patches in the NixOS version that help with that.
You probably won’t get any native NixOS things to execute under flatpak without some heavy hacks, since they don’t expect libraries to be where they are in the flatpak environment. I’d guess this is why your manually added binary doesn’t work.
Have you installed steam with programs.steam.enable? The differences between “native” and flatpak versions of steam should be very minimal, since steam bootstraps the same runtime in either case. There’s virtually no difference in the actually running code, except for the sandboxing. Fully deleting the steam directory so it properly re-bootstraps itself might help.