Previously in Epiphany/Web, you could launch different profiles in new instances using the --profile flag, similar to Firefox. This was handy if, for example, you wanted a separate instance using a different Google/Firefox Sync account. But everything changed when
the Fire Nation attacked elementary switched to Flatpaks, and this doesn't work any more.
apt and doing
epiphany-browser --profile="/home/USER/.local/share/epiphany/PROFILE" works. However, using the preinstalled version with
flatpak run org.gnome.Epiphany --profile="/home/USER/.local/share/epiphany/PROFILE" (or the longer version I found in the .desktop file,
/usr/bin/flatpak run --branch=daily --arch=x86_64 --command=epiphany org.gnome.Epiphany) doesn't work; it just launches an incognito window.
Is there a way to get profiles to work in the Flatpak version of Epiphany/Web? Do they need to be stored in any particular place?
Figured it out myself, the solution is to go into System Settings→Applications→Permissions and give Gnome Web access to Home & System folders! Leaving this here in case anyone else has this problem. As per Michael C.'s answer, it only needs access to Home, and profiles are stored in