FWIW Bazzite, SteamOS and CachyOS (Handheld Edition) are essentially all doing a very similar thing; bundling a load of packages like the kernel, Valve's Gamescope compositor (lightweight, gaming mode), Proton software layer for emulating Windows code etc into a one install-able image. SteamOS is typically older versions of those parts and has less stuff included, where as Bazzite & CachyOS include more software by default. Bazzite has a slight difference in terms of being build on Fedora Linux rather than Arch Linux like SteamOS & CachyOS.
Personally I've found SteamOS to be faster in-terms of installing and start-up, but they end-up all doing the same job and looking the same. If youi're only ever downloading Steam games and don't intend to do anything else then SteamOS is arguably good enough to use on non-Valve hardware now.
In terms of hardware if you have an AMD, or Intel build you should have a relatively good time with any of them although there are still things like WiFi cards, or bugs that can catch you out. Nvidia isn't supported in Gamescope and therefore you can't currently install without issues. Bazzite warns you as much on the download pages.
Any of them largely work very well with just a controller, but you still get games which have annoying UI, or launchers that can break the controller experience. Borderlands 2 is an example where the splash screen that you can't navigate past with just an XB controller. You need trackpads, or keyboard and mouse to type '-nolauncher' into the properties. Stuff like this unfortunately still happens from time-to-time so every so often you might need to grab a kb&m, or have a small keyboard like the Logitech K400r around in-case.
