What is the value of $XDG_DATA_DIRS environment variable? Does it include paths for flatpaks installed globally and for flatpaks installed per user before the host system paths, where the system version of epiphany is installed?
$ echo $XDG_DATA_DIRS
/usr/share/librem5:/usr/share/librem5:/home/purism/.local/share/flatpak/exports/share:/var/lib/flatpak/exports/share:/usr/local/share/:/usr/share/
$ ls /usr/share/applications/ | grep Epiphany
org.gnome.Epiphany.desktop
You could also add a custom path to $XDG_DATA_DIRS, for example /home/purism/.local/share/, and choose which .desktop files to add to application directory there if you do not want all flatpak applications to have precedence.