It's an unofficial repository (PPA) for Nvidia drivers on Ubuntu and Mint: https://launchpad.net/~graphics-drivers/+archive/ubuntu/ppa
If you add "ppa:graphics-drivers/ppa" in "Software Sources", you'll be able to install newer driver versions in the "Driver Manager". For a 50-series GPU, you'll want at least version 570 IIRC.
I think those should be fine with Mint 22. You'll just need to use the graphics-driver-ppa to get an up-to-date Nvidia driver.
So, it's basically up to you if you want to play around with another distro or not. But tbh it sounds like Mint is a good fit for you.
What are the specs of your new computer? Mint can struggle with brand-new hardware (e.g. new GPUs from AMD/Intel). Or did you purchase a new PC that officially supports Linux (Mint)?
True, Linux applications (e.g. apt, dnf, pip, but also rm, sudo, and many more) would be more precise.
For Arch, it's probably not so easy to define "essential" packages, as it, for example, supports many different bootloaders. It is of course also a question of distro philosophy and target audience. Personally, I've noticed that "rm -r" as root prompts for every file on RHEL but does not on Arch...
E: Removing essential system-critical packages is not permitted. This might break the system.
You can still do it if you really want, but even Linux rightly has some protections against breaking your system.
I'd say Mint is fine for gaming, as long as your hardware is supported. I'm using it with an Nvidia GPU on X11 and I can play all the games I want to play (Steam is Steam after all). My main gripe is that multi-monitor VRR doesn't work on X11, but it hasn't pushed me to another distro just yet...
For people/beginners that mostly want to game on a computer, I'd say that actually something "immutable" like Bazzite might be one of the best options.
Both KDE Discover and Gnome Software offer similar functionality. You should also be able to use them without their respective shells.
The PPA just provides the packages, you can actually install them through the Driver Manager after adding the PPA. However, without the PPA, the newest available version seems to be 550, which is not new enough for a 50-series GPU.