Inexplicably, now it's decided to start working right. I didn't do anything to the existing backup job.
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I'm glad it seems to be working now but I wish I understood why it wasn't working previously. That makes me nervous that it could stop working in the future without notice.
The only real difference is that I set up a brand new set of VMs to be backed up in the same way to the same remotes using two new backup plans. I guess this is kinda solved...