Flatpak apps cannot set their own permissions “on installation”. If flatpak tells you some weather app uses only the network permission then that is all the app is going to get.
For an app to be able to change its own permissions, it first needs permission to the flatpak overrides directory. Any app that does this gets an “Unsafe” designation in gnome-software.
Also about most apps requiring filesystem access to work: I have 41 flatpak apps on my system (Silverblue so everything is flatpak). Only 6 have access to my home or Documents directory. (11 apps requested full filesystem or homedir permission, but 5 of these work perfectly fine after I turned off their permissions in Flatseal).
Notably, “large attack surface” apps like Thunderbird or Firefox don’t have access to my Documents. File uploads and email attachments go through the file picker portals.