I have not found a way to start the game directly from game mode in Steam. Adding Madden NFL 22 directly as a non-steam game gives an error off missing EA Origin services.

That is why I chose to only add EA Origin to my Steam library and click on the game in the Origin launcher. Everything works great except for occasional loss of network connection (restarting from EA Origin launcher helps).

Exiting Madden NFL 22 works like a charm. But by the love of god.. every time I close a non-steam game, in this case the Origin launcher, the loading icon from Valve stays on top - have to 'abort' it with button B and then I am back in my library.

Play time is not recognised in the Steam library but I guess I have been playing for about 15 hours the last couple of days.