Unfortunately the container breaks certain extensions (e.g. keepassxc-browser). And that's ignoring the fact that, at least on my machine, the flatpak fonts look terrible for some reason and I often run into issues with the Gnome theme not being properly inherited.
Try flatseal and see if you can tweak the permissions.
Re: terrible look, I face the same thing with Telegram (the mouse reverts to whatever is the default in Qt and ignores the system settings). I haven't been able to fix that.
When installing distro, I usually remove some included packages and install newest Firefox, LibreOffice, Gimp, Inkscape etc from Flatpak, because Flatpak has newest versions. Many packages do not update .deb anymore, they have moved to Flatpak.
I've been using the Debian packages supplied by Ubuntuzilla (despite their being hosted on gag Sourceforge): https://sourceforge.net/projects/ubuntuzilla/