You might have just had bad timing - the release notes for Proton 8.0-4 mention
Fixed EA Desktop having a blank log-in window after its recent update.
It's currently a test release so should be out shortly, but if you want to use it immediately there's some info on enabling it early at https://www.gamingonlinux.com/2023/09/proton-80-4-is-now-in-testing-with-a-release-candidate/
I don't find the Steam Deck unwieldy. It is big, but the design and weight distribution makes it comfortable to hold - I don't even think about it when I get into a game on it. For comparison, I find the Switch Pro controllers very slightly smaller than I would prefer, but comfortable too, and the DualSense to be about right.
Steam Deck doesn't currently support being the streaming host (at least, SteamOS currently doesn't - but I wouldn't recommend wiping that as it's one of the best parts of the deck).
it doesn’t support any anticheat for any major games
That's pretty strongly worded - I know Bungie and Epic (Fortnite only) have both refused, but I consider quite a few of the many games that do support anti-cheat on Proton to be major https://areweanticheatyet.com/table/1/?search=&sortOrder=&sortBy=status