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 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.