Thank you; this is what fixed it for me on Steam Deck.
To any others with this issue on the Deck: I went into Desktop Mode, opened Steam, started MSFS 2024, then when I got to the unresponsive green login button, I pressed Alt+Enter (I happened to be docked at the time). The game switched from full screen to a window, and the button then turned white when clicked, and I could log in. Once I was logged in once, future starts did not prompt me, and I could run it in Game Mode like normal.