An update: the (very) good news is that my Backblaze backup is now complete and nothing had in fact been wiped. So I have now purchased it.
What happened is that when I followed their instructions on the 2nd computer to inherit the backup state of the 1st computer, the backup client application erroneously indicated that it was performing an initial backup, as if it was starting from scratch. Apparently it's not supposed to do that, but it sometimes does, according to their tech-support.
Compounding the confusion, the inherit backup operation renamed the backup set on the Backblaze servers with a new name and made all the previously backed up files disappear from view. Neither the application nor the website provided any indication that it was performing any de-duplication, whereas behind the scenes that is in fact what it was doing.
It took days of emailing back and forth with tech support to figure all this out. At one point they asserted I'd never completed the initial backup and that's why it showed I still had more than 2 TB to upload; I knew I had done that upload and assumed either their system was broken or I was not understanding something.
One thing is for sure: none of this stuff is simple. Illustrating the complexity of the problem, when I write the "the previously backed up files disappear from view" on the website, that's true, they did. However you can go "back in time" and they're there. When you complete the backup from the new computer, they reappear "in the present", rather than only in the past. There is a certain logic to that, for sure, especially considering files can move location between one computer and the next.
However IMHO the UI on the website must do better in providing feedback to the user as to the status of their files. And it certainly their client application should never display that it's performing an initial backup when it's not in fact doing that. I'd also like to see the client application indicate when it's de-duplicating files rather than re-uploading them.
On a considerably more positive note, one good thing is that their client never thrashed the disk, crashed, or consumed excessive amounts CPU, in total contrast to the shockingly awful Amazon client application. So while the UI needs work done on it, at least the core file transfer functionality works.