The Android private DNS setting is just for a DNS-over-TLS resolver. The only thing about it that's private is your queries are encrypted en route to the server (traditonal DNS is cleartext). There's no filtering or blocking.
Some Android versions also have a hard coded DNS server set to Google, which based on my tinkering uses DNS-over-HTTPS. Not only is it annoying but I find it awfully insecure - even if you think you have stuff locked down it might just not be. I fixed that issue by blocking all DNS-over-HTTPS servers in my router, and also have all outgoing requests to port 53 redirected to my local resolvers (Pihole + Unbound).