Well, exactly as I described. I had a K8S node that was cordoned when Longhorn was installed, so Longhorn saw that node but did not enable it, and no instance-manager-e-xxx pod was created on that node. In this state the ability to backup a volume (either from the Volumes tab or the Backups tab) is disabled because of a "instance-manager-e-xxx pod not present" error. If the node is then uncordoned the pod is created and backups are OK again.
Interestingly it seems that if the node is then disabled in Longhorn the pod is not removed and backups are still OK. So maybe this only applies when installing Longhorn to a cluster where a node is cordoned.