> I'm having trouble understanding how else this is supposed to be? I understand that live migration is a thing, but even in those cases, a VM is "hardwired" to some physical server, no?

They mean the storage part. If your VM's storage(state) is on one server and that server dies, you have to restore from backup. If your VM's storage is on remote shared storage mounted to that server and the server dies, your VM can be restarted elsewhere that has access to that shared storage.

In AWS land it's the difference between instance store (local to a server) and EBS (remote, attached locally).

There's a tradeoff in that shared storage will be slightly slower due to having to traverse networking, and it's harder to manage properly; but the reliability gain is massive.