I would go for a pre-built computer with an RX 9060 XT (preferably with 16 GB VRAM). That GPU will run loops around a Steam machine. You can get such a pre-built for around 1000 CHF:
Interesting. To me, they both offer a desktop experience (panel, start menu, tray icons, windows, tiling, virtual desktops, …) that captures all the things I liked about the windows interface. What sets them apart for you? I get the wish for Mint KDE though, would be nice :)
If you’re looking for Mint with KDE Plasma, Kubuntu should be a fine choice. I would also not overthink it too much, it’s quite easy to tune things (e.g. remove snaps on Ubuntu) or switch distro afterwards. Mint, Kubuntu, and Fedora all allow you to play around and find out what you like (and dislike).
In my experience it works with some games but doesn’t with others. And if it breaks, the game just stops without any error message/information (you can get some info on the terminal). I would avoid it if at all possible. NTFS support is also overall still rather poor (e.g. my Mint install recently stopped working with NTFS drive until I ran chkdsk on Windows).
I think most distros would work for that usecase. I would recommend Linux Mint, Fedora (although Nvidia drivers are more annoying there), or (K)Ubuntu. Have a look at desktop environments too, as they are very important for your experience. Some of the major ones are Cinnamon, KDE Plasma, and Gnome. Mint works best with Cinnamon, Fedora and Ubuntu let you choose which ISO you want to download.
A regular pre-built with a 9060 XT 8GB can be had for about 10% less than the Steam Machine (900 Euro). That will be quite a bit faster and also upgradeable. The main advantage of the Steam Machine is the form factor and power consumption.