Plug in controller (mine is a Nintendo Switch Pro controller based on the Gamecube made by PDP).
Launch Steam.
Add Madden 19 to Steam's Game Library by going to the Games > Add Non-Steam Game To Your Library option.
Quit Origin.
Launch Steam's Big Picture mode.
Go to Settings > Controller Configuration and enable Nintendo Switch controller support, whatever else is appropriate for your controller.
Map the buttons by pressing A or clicking on the controller in the Steam controller settings area.
Go back to the Big Picture library area and check the Madden 19 controller options to make sure they are set up properly. You will not want Desktop Configuration enabled.
Launch Madden 19 directly from Steam Big Picture mode (sometimes takes 2 or 3 clicks). Keep attempting to launch Madden 19 through Steam, even if Origin protests or says it cannot connect to EA.
Once the game launches, your controller should be exclusively controlled by Steam, and you can now connect to EA.