In proper IT tradition... as soon as the new server was up and running and settled down, Hetzner got the issue with the previous node solved.
Because of course.
The little "good sysadmin" guy on my left shoulder is chastising me for not having failover and a floating IP already set up for this kind of thing.
The little "devops burnout" guy on my right shoulder is chuckling at me and whispering something about just writing a BASH script to do it in a janky way and forget about it.
I am caught in the middle and will probably do something with haproxy and keepalived. Such is the life of Ops.