So, while not absolutely the answer, it was enough to put me in the right direction, so, full marks.
The VSC server was properly registered, but someone else had gone through behind the scenes on the AD server and pulled the VSC service account out of the VMware admin group. So, effectively, it wasn't an admin anymore, and here we were.
I really wish there were better privilege definitions for VSC. Like, being able to give an account just enough privs to do its job. The Backup/Recovery defaults to using the admin user to do backups, but, with some research you can find that SMVI only needs a subset of privs, and you can define a role just for it. But then you have to set that user on every backup job. That's probably another bug.
Anyway! Thanks for the pointer.