Appimage run everywhere

As long as your glibc is compatible.

Getting flatpacks to run on a Linux distribution that is not explicitly supporting them is quite a hassle.

I’d struggle to think of a distribution that wouldn’t work (https://flatpak.org/setup/). Anyways at that point you’d be lucky to have video drivers anyway so games are probably not an interest. And someone on a distribution like that wouldn’t be unfamiliar with a bit of hassle compiling software.

There are also systems to provide repositories for appimages (and various update mechanisms)

Yeah, systems that almost no appimage implements. Welcome to version hell.

Flatpacks are over-engineered and especially when it comes to games have all sorts of edge-cases that severely degrade the experience of players.

I disagree.

Last but not least, no amount of sand-boxing is going to solve the problem that such an auto-updater

Flatpak does not auto update. Gnome software auto updates (and can be told not to).

But this is better dealt with at the repository server level with strict checks of what can be included or not (f-droid like).

Which flatpak can provide. It seems most of your issues might be with flathub, not flatpak.