Steam Deck users look here!
If you want controller support through the Steam Deck, press the STEAM button when in the game, navigate to "Controller Settings", then choose the standard "Gamepad with Joystick" preset out of Steam Input's own list. Basically just make sure Steam Input is properly ON. Then quit and restart.
For some reason, this makes the game suddenly recognize that a controller is "connected" and the controller options open up in the main menu.

If you want to make your own button config in-game after that, you can do that too I'm sure.
Also don't forget to set your resolution in the video launch settings before you play!
(I recommend 720p instead of 800p for this game on the Deck.)

- - -

With that out of the way, the game launches fine and looks great on the Steam Deck. No tweaking necessary so far apart from the video settings, which is the only thing you need to navigate with the cursor one time.
Will update the review more once I've finished the prologue, but so far so good!

EDIT: The prologue and everything around it ran absolutely fine, with no issues to performance or anything like that on the Deck so far!
I'm impressed by how well it holds up, and I can't wait to play through it all on the go, just like the PS3 times. Nobody drinks alone in Paris.

Thank you EA; This was a rare but terrific move on your part!
Keep bringing more classic games over to Steam and I'll buy them too, especially knowing some of them work on the Deck.