Is your Steam installed via Flatpak? If your Steam is Flatpak’d then it might be pulling in some extra Flatpak dependencies (e.g. Nvidia drivers) that Bottles could be using. If it’s not installed by Flatpak then I’m puzzled, because Bottles shouldn’t be using the base system for anything.
At the very least I would try uninstalling and reinstalling Steam to confirm that it is indeed what is causing it to stop/start working, and also to check if any other dependencies are coming and going alongside Steam.