So it turns out that the "corrupt" message could be very inaccurate, at least on OS X.
In my case, I had previously moved my backup to another location. While Apple says don't do that, there are many websites that show you how once you find your backup as described here: Locate backups of your iPhone, iPad, and iPod touch - Apple Support
The instructions are to symlink the moved folder back into the MobileSync directory.
In my situation, some time after symlinking the backup, I had migrated my system (including iTunes) to a new hard disk. My symlink was then pointing to a nonexistent directory!
Trying to backup, iTunes would spin and look like it was doing something for 30 seconds and then come back with the fact that the, nonexistent, backup was corrupt! That was painfully misleading.
But, once I retraced my earlier steps, I was able to find the original backup and then repoint the symlink to the correct place.
And once that was done, the backups reappeared in the iTunes Preferences list and the backups started working again! Why? Because they weren't corrupt in the first place, they were simply missing via the bad symlink!
iTunes should have immediately complained that the files were missing. That would have made more sense. But it's an unsupported "feature" so, it's on me.
At least I can help others who've run into this problem and haven't found the root cause!
Good luck.
