It boots into Game scope (as others said, it's not "big picture mode" it's a compositor stack tailored for high gaming performance) but it's nothing more than an immutable arch distribution (and immutability can be disabled for tweaking) so you could definitely swap the defaults if someone has documented how.
Would be a nice feature to have once SteamOS becomes independent of the deck.
Even the stuff in the "immutable" section isn't necessarily wiped, it's more that there is a strong chance changes may be overwritten. I'd definitely make no guarantees to anything stored on the immutable sections of SteamOS.