With tailscale, hostnames resolved correctly, but ping did not respond and SSH was impossible.

In the end, re-registering the host fixed the issue.

Since it is registered as a new device, the old one must be removed. If needed, the new device should then be renamed.

sudo systemctl stop tailscaled
sudo rm /var/lib/tailscale/tailscaled.state
sudo systemctl start tailscaled
sudo tailscale up --authkey=...