vandalazzo: i have a generic gamepad from Trust (yes, I already know it's a crap brand) and it doesn't work with Bloodstained, not even after setting it on steam as a non-steam-game.
yet, the gamepad works with dead cells (gog) just fine
This is because Dead Cells uses SDL2 with database of controllers from here:
https://github.com/gabomdq/SDL_GameControllerDB/blob/master/gamecontrollerdb.txt At least two Trust controllers are listed so if either one is yours, game knows which one to use and it just works.
Many games use this from Danganronpa, SteamWorld series, Celeste, Titan Souls and Baba is you, so even if it is shitty third party controller, you should still be able to utilize it with massive amount of titles, from GOG and elsewhere, without any setting up.
Also if there's no controller support in game, simply adding it on steam does nothing. You either have to focre the steam input support for that non-steam game (right click -> properties -> game specific steam input -> force on) or set it up and/or enable it globally for all steam games (open big picture mode from top right -> open settings -> controller settings -> select recognized controller to set it up and check generic controller support to enable support).
Also SDL2 is just one way of getting user friendly controller support in games. It does have drawbacks like rumble support, hence why having Dualshock 4 support specifically is good so it can utilize controllers haptic feedback, lightbar, touchpad, etc. without issues.