Where did you get the game from? Most platforms like steam don’t actually sell games. You pay to rent the game for life, which is different. Those platforms can change the terms of service for your rental whenever they like. If you prefer to own your games you need to visit actual shops, like GOG.
Especially as a new linux convert I would say wait a bit before switching to an immutable distro. They have their advantages but the concept probably feels pretty alien for most windows users, where you can install whatever and it just works. I’ve been a long time Nobara user and it’s pretty good, especially considering it’s basically just one guy maintaining it. But it does break on updates from time to time. My personal recommendation after getting used to linux would be opensuse tumbleweed. It’s constantly updated but never breaks. You generally don’t need a gaming centric distro, especially if you’re rocking an AMD card.