It sounds like the registration process failed during your upgrade - the authentication from VSC is checked each 10seconds, and if it fails or something has changed, it tries to re-register with the vCenter server, which is what is happening in your case - this is by design in that it should be checking, but not by design as in you shouldn't have so many sessions.
I'd recommend completing the registration again - try this:
Log into the VSC server over RDP using an administrator account
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.
Lets go ahead and start a new thread for you IT, it sounds like you might be having a different problem. If we can't figure it out over the new thread, we'll just get you to open a new support case and work through reviewing the logs in more depth to find the RCA