I want to play my EA Steam games with the Steam overlay

As @fultpl mentioned in my bug report thread on the issue, it seems that the Steam overlay now only works if the EA overlay is enabled as well in the EA app. So go into the EA desktop app, click the 3 bars in the top left (some call it a "hamburger menu") and go to settings, then go to "Application" and turn on the overlay option near the bottom. Personally I turned off some in-game notifications just above that option so I hopefully don't get bothered by them in-game.