Almost any machine should do okay for this general purpose.

Almost any machine should do okay unless the user is easily annoyed by fan noises.

The DeskMini is slightly larger and heavier than most mini PCs, but it is the only one I know whose cooling fan can be replace with a high-quality quiet Noctua fan.

I have run Fedora on ThinkPads for years but just got a Mini PC this summer to be a TV and audio/Roon server- ThinkCentre M80q. It's great, drives a 4k display, all the things work.

look into geekom, they seem to have a good price to specs ratio

There's a "HP EliteDesk 800 G2 Tiny i7-6700 3.4GHz" that can be had on ebay for around $200, that's a pretty powerful little box. My son had no problems using it for light gaming. Note that these can get a little toasty under long, heavy use and throttle, just because of the size. My one that does camera processing seems to need to run without the case on.

I've been using the Beelink SER5 with Manjaro Linux since a month. It also comes preinstalled with Linux, but I got the Windows version. Runs smooth and perfect, no issues at all except for WIFI after suspend that had a small patch. Definitely recommend it. Much better than a laptop since portability isn't the main goal as I prefer a big monitor and use it as a daily use PC. For around $400, glad to have this little beast with 32GB RAM and Ryzen 5600H.

Beats buying a laptop for double the price and likely worse linux compatibility.

As far as Mini PCs go, I'd consider Intel NUCs (currently NUC 12 Pro line), primarily because they have great firmware lifecycle support. A lot of cheaper brands in the NUC-like Mini PC space don't consistently release firmware updates (if at all), e.g. to fix security vulns, which is a deal breaker for me. Intel NUCs are validated for Ubuntu and RHEL. The main downside is that you'd be relying on an iGPU for gaming, so do some research on whether the Core i5-1240P's Xe iGPU would be able to handle the games you're interested in.

I've been very happy with the Asus PN 50/51 which I use to run my not smart TV. They run very cool and quiet despite having a fan. Choose the level of AMD processor that suits your needs. All that and a fair bit cheaper than the Intel NUC's.

You could consider getting a steam deck and using it docked. (nvm just reread and don't know how well Fedora would work, but it would satisfy the power and smallness factors pretty well and be portable to boot)