I would not use mint with cinnamon if you intend to game. It has some compositing issues which can cause performance problems. I personally ditched mint for Manjaro KDE, but you could give something with gnome a go, too. PopOS is popular, while for KDE, Neon seems good. My non-techy sister happily uses Vanilla OS.
Also, the open source nvidia driver is not suitable for gaming. As explained by others, it leaves much to be desired in terms of performance.
When the display is not detected, I assume it’s not even showing up in display settings? When I turn on my projector, I have to “enable” it in display setup before I get output.