Update: December 10, 2017
My original plans were to hook this failed Pipeline Drive up to my new LGA1151 computer and to use Periform Recuva to see if I could pull any of the files off of it and onto a WD My Passport HD. Otherwise, sending it to a place that specializes in fixing broken HD's.
Well...
After getting Windows 10 Home 64 installed and working, I hooked up the Pipeline. On booting, I got some messages stating that the Pipeline was being scanned for errors.
After a few minutes, I'm looking at a "working" Pipeline HD!
I quickly grabbed the WD My Passports HD and copied over the entire contents of the Pipeline, one directory at a time.
Only lost 2 files: one HD movie, and one TV show.
I don't know what W10 did that W7 couldn't do, but I got all my video files back except for two.
Scanning the Pipeline HD, I am still seeing the same SMART errors that I posted a photo of here earlier.
As I can no longer trust this HD not to fail again in the future, I have pulled it from the system and it will just be an emergency back-up in the event that I need any of the files on it. I don't see any sense in spending a lot of money on it when I can simply buy a new HD for probably much less than it would cost to fix this one.