I think you still have to specify the URL up to the greader.php page, maybe in your case it would be

https://freshrss.example.com/api/greader.php

I set this up on my instance about a week ago and it works perfectly, thank you!

I recommend NSD or Knot for strictly authoritative servers. BIND is great too, but it is built to do both authoritative and caching DNS which makes it a bit too “big” for the task of serving only authoritative DNS data. You can definitely configure BIND to only serve authoritative data though.

I can’t comment on running from a container, I’ve always worked with NSD/Knot/BIND building directly from source.