I have mine only internal so i haven't ran into that. But check console. You mention mobile so if you're on android you can hook it up to your pc and use debugging through chrome.. In the past I've had success looking at error messages to see why my requests were failing. Usually because i wasn't passing headers correctly.
I use symfonium and it looks like it let's you pass custom headers if needed. Good luck
I didn't have "deeply rooted" projects, but i started by just creating new projects on codeberg. It gave me time to test it out before i moved the rest of my projects which i did ~6 months later.
For your use case i would consider the same. Figure out where you might like to migrate to and give it a test run with new projects. Then decide if you want to migrate the deeply rooted project(s) or not.
I didn't have to worry about the community much, both because i didn't have large projects and also because i was on gitlab to begin with. If anything going from gitlab to codeberg i had the same if not more engagement with my small projects. But my motivation for my migration was just wanting to use services that support what i support. For me that's what was most important in the long run.
Also re: github container registry. Forgejo (what codeberg uses) supports this: https://forgejo.org/docs/latest/user/packages/container/ . These images would then also show up on your project page under "packages" tab if it has been enabled