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.
From my understanding, Flatpak is built on top of OSTree, which will automatically deduplicate files across different packages. That said, I’m not sure if this extends to downloading packages. The site claims that it does do “delta updates,” which would hopefully mean that it doesn’t download files that are already on the system, even if they’re part of another package.
I’m just going off what I read in the docs. Someone with more understanding of the system can clarify.