This appears to be a conflict between systemd-resolved and resolvconf.
I'm able to recreate the issue with the following steps
1) Fresh 20.04 install
2) boot to friendly recovery
a) enable networking, drop to root shell
b) name resolution works as expected
3) reboot to fully booted system and install resolvconf
4) boot to friendly recovery
a) enable networking, drop to root shell
b) name resolution no longer works
c) /etc/resolv.conf is empty
d) restarting systemd-resolved populates /etc/resolv.conf and name resolution
begins working again
This appears to be a conflict between systemd-resolved and resolvconf.
I'm able to recreate the issue with the following steps
1) Fresh 20.04 install
2) boot to friendly recovery
a) enable networking, drop to root shell
b) name resolution works as expected
3) reboot to fully booted system and install resolvconf
4) boot to friendly recovery
a) enable networking, drop to root shell
b) name resolution no longer works
c) /etc/resolv.conf is empty
d) restarting systemd-resolved populates /etc/resolv.conf and name resolution
begins working again