I've tried again with the official new big picture client that came out today, pretty much have the same issues still.
It's also worth mentioning that The Long Dark consistently starts but is stuck behind the swirling steam background. It flashes up once very briefly and then you can't see anything further. Also worth noting that when this happens (always for this game) pressing the steam menu button to kill the game doesn't bring up the overlay that lets you stop the game, although the buttons "work" since you can still kill it by counting keystrokes. At any rate this one seems to be a pathological case for the issue.
proton 7.0-5x1 2023-01
Your system information
Summary
When I launch some games using the new big picture mode (the steam client is in BETA, so "-tenfoot" starts in new big picture), I notice that there is flashing between the swirling Steam icon that is presented when a game is launched, and the game splash screen. It doesn't do this on every game, but there are a few where it does it all the time, such as Gauntlet and Supraland (both run on Proton 7.0-5). Others such as Subnautica seem to work without a hitch.
When I turn off the BETA and go back to the old big picture mode, I dont have any of these type of problems.
Usually the flashing can be cleared up by pushing the "XBox" button in the middle of the gamepad to bring up the Steam interface, and then going back to the game.
I have a setup similar to SteamOS 2, where when I start the user session it uses the latest versions of "steamos-session" and "steamcompmgr" from SteamOS 2 (basically it doesn't start a desktop). The binaries for both are pulled directly from the Valve SteamOS 2 repositories.
Please note the one change I made to "steamos-session" was to start the Steam client in "tenfoot" mode rather than "steamos" mode (that's so I can exit the program, I have a gamepad-friendly login greeter that I use to switch between different users which lets me keep games available per-user separate, a thing the current multi-login steam client doesnt seem to handle well).
For whatever its worth (its hard to know how this thing is implemented), it looks a lot like the "swirly steam"-thing and the actual app are fighting for top-level focus, as if each was a separate full-screen window of some kind, and the system wasn't able to sort out which to present on top. At the risk of being presumptuous it might be worth looking at how that sort of thing is handled in the old big picture mode compared to the new one, there could be some clues there. Maybe the new big picture client is assuming some functionality in a desktop window manager that isn't in steamcompmgr?
Thanks,
jonbitzen