I could be wrong here, but perhaps those game "updates" are just adding profiles specific for those games into GeForce Experience/Adrenalin. I've never needed a game specific update driver for Nvidia, and from what I've heard on the AMD side of things - generally you don't either. Every now and then a new Mesa build might make things work for a game better however, which is why they mentioned the distro (although I think the Steam Flatpak tends to include the newest Mesa anyways).

Since neither manufacturer has a dashboard for Linux, there are no updates needed.

Most game support related updates come from Proton/WINE/etc on the Linux side of things, not the drivers in my experience. Have a look at the Proton release notes, and you'll see game specific updates generally landing there

I'm mentioning the distro as Arch has a very different update path than Debian. Different package managers handle drivers differently. Also there are different Security decisions between say Rocky and Mandriva. All of this will affect drivers. Linux is just the kernel your distro is your OS it's not just a DE there are a lot of decisions being made that aren't just GUI.

Valve uses Arch for a reason keep to that. Stay up with Proton reporting and update your whole system regularly and read patch notes. However if you want the latest and most stable experience just mirror whatever Valve is calling Steam OS these days

For Arch and AMD you can just use the AUR as it'll grab whatever is best for your setup. Most linux games and bug fixes are focused on the community drivers. However if you're seeing somewhere that the proprietary drivers will give you benefit you can install the amdgpu_pro driver.

https://wiki.archlinux.org/title/AMDGPU_PRO