I know Steam supports controllers, but does lutris and Heroic Games Launcher support it too? (PS, XBOX, SWITCH)
Yes, all normal controllers should work out of the box on a modern kernel.
That includes official Playstation, Xbox and Switch controllers.
Like kernel 6.1?
Yes, 6.1 is recent enough to have all drivers for the controllers mentioned above.
As long as the game controller shows up in Linux, it will work in Lutris as long as the game supports controllers.
Everyone else covered the main details, I’ll just add if you want to use an Xbox series X controller I found the Microsoft wireless dongle works best for wireless usage.
You would need to install the third-party xone kernel module though.
Games from Steam will work with just about any controller, non-steam games can usually be made to work by launching them from within Steam, so add the games to Steam when installing through Heroic or Lutris eg.
One possible issue is that mixing rpm & flatpak runtimes can cause issue’s with Steam Input, atleast it did for me on Fedora 38 where Heroic installed as a flatpak couldn’t interact with Steam Input where Steam was installed from rpm.