Valve says your global Proton setting is the last thing Steam looks at
Valve has written down the order Steam uses to choose a compatibility tool, and the global default in Settings comes last. A Valve developer posted the chain on October 14 on Valve's SteamOS tracker: "Per-game override > Valve manual pin > Steam Deck compatibility config > Linux native > Global compatibility tool config."

Valve says a desktop SteamOS install that runs Steam the way a Steam Deck does behaves the same way. Image: Valve Image: Valve
The reply closed issue 2932, filed the same day by an owner who set the default compatibility tool to something other than Proton Stable, restarted, and then found every game without custom settings still showing Default as Proton Stable. Valve's answer was that this is "the current expected behavior on the Steam Deck".
So this is expected behaviour rather than a bug, and the global setting is the last fallback of five.
Valve says a desktop SteamOS install behaves the same
The reporter is not on Valve hardware. They run SteamOS 3.8.28 on a desktop PC, with no Steam client beta and no SteamOS beta, which is the obvious objection to an answer about a Steam Deck.
Valve's follow-up eight minutes later covers it: "If you're running a system install that's running Steam similarly to a Steam Deck, I'd expect the same behavior." Valve's own wording on the list is "should be something like", so this is a Valve developer's account of the expected priority rather than published documentation. It is also the only public statement of the order.
A per-game override is the one setting nothing outranks
Valve's per-title ratings and its manual pins both sit above the global default. So on a library of games Valve has classified for Steam Deck or Steam Machine, the tool you picked in Settings is consulted for whatever is left over.
If you want a specific Proton build on a specific game, the per-game override in that game's Properties is the only entry in the list that nothing above it can overtake. A game Steam considers Linux native also outranks the global default, which is why a native build keeps running natively however the default is set.
A report filed two days earlier shows the cost of a stale per-game setting. A Debian user found several titles they had played the previous month greyed out with a "Windows only" note, and Valve read it as a game "configured in Steam to use an invalid or removed compatibility tool". Forcing a Proton version per game fixed it, the reporter said.
That was Valve's reading of one report, and the logs it asked for were never supplied, so it is not an explanation of every greyed-out download. Both issues are now closed.
Steam's defaults have been shifting around this. The beta client started running non-Steam Windows binaries through Proton on its own on October 10, and Proton's stable branch moved for the first time in 49 days on October 12. The global default still exists, and it is the last thing Steam asks.