I think gamescope needs a Wayland session, as it renders games in their own Xwayland "screen". Mint does not use the Wayland session by default, but the old X11 one. I think Mint added an "experimental" Wayland session recently, so you could try to enable that.

Debian based distros tend to be slower to update for stability reasons so your graphics drivers might be out of date

https://github.com/ValveSoftware/gamescope

AMD requires Mesa 20.3+, Intel requires Mesa 21.2+. For NVIDIA's proprietary driver, version 515.43.04+ is required (make sure the nvidia-drm.modeset=1 kernel parameter is set).

Do these games launch when the gamescope option is disabled?

You can also check the logs by selecting the game in lutris and pressing the arrow beside the play button on the bottom to bring up the logs and see if theres any useful info there

I did not know gamescope was a flatpak... For me gamescope did not do anything after installaling for like 2 hours of troubleshooting why it was not installing. I'm also on mint