Just for future readers: the docker[-compose] required rebuilding of the container so the new volume is picked up:
docker-compose up --detach --remove-orphans --build pihole
After that the new /etc/resolv.conf was in place.
Though I also want to voice disappointment here, this should have been the default and as a programmer I am so burned out that I don't even wanna know why is it not the default.
By default, Docker container's would use DNS servers as defined in the host system's /etc/resolv.conf, either directly by copying the file into a container or indirectly via Docker's internal DNS resolver using the nameservers from /etc/resolv.conf as upstreams, depending on a container's network mode.