It is not that lookups are not working, which they are. If you look closely the CoreDNS plugin is failing install.
Error on install CoreDNS plugin
So, my question was around a potential service conflict.
I am guessing you are focussed around the no-internet log entry, which I think is a by-product of CoreDNS not being there.
I hope this helps.
The hassio-supervisor container is unable to access localnetwork DNS server answering on host IP address aka Pihole. Provided below is my reasoning why I think so. Happy to be proven wrong if you have additional info.
-Tried with PiHole but pointed /etc/resolv.conf to an host PiHole DNS server — Did not work
-Tried with PiHole but pointed /etc/resolv.conf to an internal DNS server which performs lookups from host PiHole server — Worked
Any dev wishes to weigh in on this?