If you have issues with your controllers with the flatpak I would try switching to the native package. There should be a .deb package for steam. If it doesn't change the behavior, then I would assume something else is causing a problem.
You can also install flatseal and try messing with the dbus setting, should be like half way down the menu for the steam flatpak. Since flatpaks are usually not given access to the entire system I could see it not having the correct permissions.
Can I ask how you have steam installed? IIRC Mint does not use snapd so is it via flatlak or apt? If it's Flatpak it could be that the flatpak doesn't have the appropriate access to dbus which can be changed with something like Warehouse or Flatseal.
I typically perfer a native .deb (or .rpm in my case) for something like steam.
I have pretty similar hardware, but I do not have an NVIDIA card. Can confirm that Bluetooth is pretty dog shit most of the time, but I generally use wired accessories (preference). I did have some pretty good luck with an ASUS bluetooth adapter which worked better than the onboard one. Controllers work great with steam both wireless (save BT issues) and wired. I also run Fedora on all my machines so I couldn't speak to anything Mint specific but I have been running Linux for a very long time and have found that some distros have better support for newer hardware than others.
Mint is fantastic and that community does great work, all systems I manage that I do not own (family) run Mint. It is what i would recommend for any new linux user. However, for your use-case it might not be a terrible idea to give something else a try (if you haven't already).
There will always be troubleshooting that needs to be done regardless of the OS, as others have said, hardware manufacturers do not take (desktop) Linux seriously unfortunately.
Good idea, then we know FOR SURE it will never get built!
Shell colors change the game really. I did this at work; when I'm on the admin server my prompt is rainbow, other servers black (I use a solarized light terminal). It has saved me so many times.
Glad you got it working! I turn that setting on for all my games and I honestly didn't even think about asking if you had it on.