I'm not sure I'd install tailscale as a flatpak even if it were possible. I've always seen flatpak as a way to install large, potentially crappy desktop applications without polluting your system. OBS studio is a perfect example - it's a great app but it's the only one I use that uses QT, thanks to flatpak I don't even have the QT libraries installed on my system.

Tailscale is more like a system service that I'd prefer a distro package for (Arch Linux repos contain Tailscale, btw).

You don't have QT libraries installed on your system. You just have then in some archive somewhere along with a bunch of copies of stuff you do already have installed on your system.