Are you sure that Steam is uninstalled completely from your filesystem. Steam creates a “.steam” user directory and leftover data in the directory may be conflicting with the flatpak. Delete it if it exists and then try restarting your computer. The flatpak version of steam sandboxes the .steam directory.
As always, Steam is proprietary software and doesn’t bother to follow freedesktop standards. The support I can give can only go so far. GOG is always a better solution as it can be used with a libre client like Heroic Games Launcher or Lutris without DRM or invasive anti features. You do not need Steam to leverage Valve’s contributions with Proton and Vulkan.