Love it when people speak with authority and are confidently incorrect. Eugenia is right.

You could potentially use flatseal to grant the flatpak the necessary permissions, and you might find out what those permissions are by looking for other users experiences with the flatpak version.

Or, you find the .deb file and it installs natively without being sandboxed. OR, you can find a PPA repository for it, load said repository and install your software.

But those things require learning a little. Linux rewards self starters who can use a search engine and forums. Hope this maybe points you in the right direction.

Yes the wallpaper stuff could be problematic.

But desktop integration works without issues. App stores also use flatpak install directly instead of packagekit, at least on cross-distro desktops like KDE and GNOME. Which works way better.

Regarding "they dont work well for system related stuff", on Unix stuff everything is a file, and especially dedicated apps like a wallpaper switcher can have very specific static filesystem permissions, allowing them to do exactly what they need.

Appimages are totally different, a flawed concept by design and have no installer by default so they often have no integration at all. They are also not sandboxed at all and thus just as unrestricted as system apps, while they have no repo, no updating mechanism, no shared libs and are basically a security nightmare.

I dont know about snaps. Their sandboxing needs Apparmor so it is not cross platform.