So it turns out that only a small number of recordings on the EHD were corrupt, and it was just a coincidence that I picked two of them to transfer over to the new Hopper's DVR as my first test. The majority of files that I rescued from the old Hopper's drive are playable from the EHD and work just as well when I transfer them from the EHD to the new Hopper's internal drive.
So, I don't think the unplayable ones were corrupted when I was moving them off of the EHD. The unplayable files must've either become corrupted while in the old Hopper, or were corrupted during the transfer onto the EHD when I had the old Hopper's drive temporarily in the new Hopper.
As for the theory that the new Hopper just needs more time and then the unplayable files will work, I'm not holding out hope. It has been several days and what's unplayable is still unplayable.
I did consider doing a drive swap in the manner suggested by the last poster, but it's not my name on the account, and I'm under orders to make sure there's no funny business. So officially, neither of the Hoppers have been opened, and we made our backups on the EHD before the old Hopper died, just like you all are doing now, right?