I don’t understand what problem they are meant to solve. If you have a FOSS piece of software, you can install it via the package manager. Or the store, which is just a frontend for the package manager. I see that they are distribution-independent, but the distro maintainers likely already know what’s compatible and what your system needs to install the software. You enjoy that benefit only through the package manager.
If your distro ships broken software because of dependency problems, you don’t need a tool like Flatpak, you need a new distro.
I have had very good luck with AppImages, and not so much with flatpak(s). The reason for the latter is, well, that I couldn’t get the flatpak-installed apps to pin to the panel on cinnamon (although that was sort of a long time ago, and I may have learned a few things since then), which means, I guess, that they don’t (or didn’t) integrate well. Then there was the memory thing, which, as @donuts@kbin.social points out should not really be an issue. I think I remember too that they were kind of hard to completely uninstall.
I could not even get snaps or flatpacks to install in Debian stable. Infrastructure too old. On Ubuntu the one non-system snap I installed kept loosing it’s data presumably when it updated. So my experience with both are quite poor.
AppImage had no issue with except I had to manually do some xommand line stuff to integrate it.