In my experience running non-steam games through steam with proton is the best way to play those games too. The only time I've ever had to use lutris was when I had to install some DLC for a GOG application on the same prefix as the game because it had a separate exe installer for that DLC. I haven't been able to figure out a way to do that through steam. But once I got that done I just ran the game through steam and it worked perfectly. The heroic games launcher gets suggested a lot too but I literally have never been able to get it to work for a single game.