Great idea for a thread. I made a list of the Steam games we like. A lot of these require basic controller remapping. For most you only need to remap the d-pad to "joystick move" and you are done. I tried to note the games that need a little more extensive remapping.
Multiplayer games
-any 1v1 fighting game. We play King of Fighters XIV the most, but also Injustice 2, Killer Instinct, Street Fighter IV (better than SF V).
-Tricky Towers (probably the game we play the most)
-Overcooked
-Bro Force
-Brawlout
-Streets of Rage 4
-Rayman Legends
-Nidhogg II
-NBA 2K Playgrounds 2 (requires some creative button remapping)
-F1 Race Stars
-Grip
Single Player games
-Steamworld Dig 2
-Hades (works surprising well with polycade controls, but you sacrifice some precision in aiming)
-Lonely Mountains Downhill
-Limbo
-Inside
-Yoku's Island Express
-Ori
-Pac Man Championship Edition
-Dead Cells
-any "bullet hell" game is going to work really well. I like Sine Mora Ex
-most racing games are great on polycade with a little remapping. I like Grip and Redout.
-The Trials (motocross) games are cool. Fusion is my favorite Trials game. The only one that doesnt work well is Trials Blood Dragon because it has some twin stick stuff going on.
-Steep (this requires a lot of remapping, but it is really fun on the polycade)