Might be worth trying gamescope on another known working game too. Would help narrow down if it's gamescope itself or the game you're trying to use it on.
Interesting. I haven't had any issues with gamescope for a long time... Back when it was new I definitely had issues running things with it... But it's been a long time since that's been the case. I can recommend running steam from a terminal and viewing the output after trying to run the game with gamescope. It might point you in the right direction.
Should be something like - "gamescope -W 1920 -H 1080 -f %command%" I will also note that mine tends to crash/not launch if I don't put Mangohud arguments BEFORE gamescope. The w and h arguments are just width and height resolution, so set those to match your monitor. The "-f" is for full screen.
In my experience this is almost always the compositor not properly syncing display frames with the game render frames. The best solution I've personally found is to run the offending game through game scope. It worked for Fallout 4 on my rx 7600xt gpu when I went to lock the fps to 60 (to avoid physics bugs, stupid Bethesda). Without game scope locking the frame rate caused horrendous stuttering despite solid fps.
Maybe not too helpful, but could point you in the right direction: you used to be able to use "gksudo" to get the graphical popup requesting your password in lieu of sudo which would only ask for a password in terminal. I believe gksudo is deprecated/non-existent at this point but there's got to be an alternative out there. Best of luck!
Weird! I'm running Bazzite kde (so fedora based, like Nobara but with different tweaks and it's atomic), and gamescope gives me zero problems. Might be some weird combination of software versions between gamescope and kde in your system. Or just a general kde config problem. I love kde, but all that customization can absolutely come at the price of stability at times.