Debian 11.3…
None of the other hints worked, only @RVital’s way…
Thanks.
This works for me, issue was registry-1.docker.io not resolving in IPv6
34.203.135.183 registry-1.docker.io
140.82.121.34 ghcr.io
Stared at it for a couple of days, searched for a solution a couple of days, but answer was right in front of me the whole time 
Anoyher and probably more viable solution than the above workaround (ip addresses can change at any time) can be found here.
Hello, Just tried the installation of home assistant supervised on Debian 11 Bullseye myself.
I tried now several times, but dns seems to fail after the home assistant supervised installation.
Somehow it will not connect to the local DNS server. (which in my case it recieves from DHCP. )
To overcome that I added now the google DNS servers (with the procedure above).
sudo nmcli connection modify "Supervisor eth0" ipv4.dns "8.8.8.8"
Remote DHCP now seems to work for the external URLs. However, my esphome boards are not reachable due to a failing dns request (The requests were working before the home assisstant supervised installation)
Checking the /etc/resolv.conf however shows the local DNS server as being configured:
$ more /etc/resolv.conf
# Generated by NetworkManager
search localdomain
nameserver 192.168.2.1
nameserver 8.8.8.8
Anybody any clue?