My wife has the same problem. The hard drive of her 2011 iMac crashed so she had a new one installed. But when she went to transfer all her old data over using Migration Assistant and Time Machine, MA would not see the drive. If I allow TM to do a complete backup of the new drive, MA will then see the drive but only the most recent backup, which contains no data. I tried dragging that last backup to the trash, but when I do, MA again does not see the drive. We took both the iMac and the backup to a Genius Bar but the technician could not solve the problem. He tried another cable then said we'd have to drag files over in the Finder manually, reinstall her apps, and reconfigure her email client, etc.
What at I find especially annoying about this is she did everything right, keeping a current backup using Apple's backup software. If you can't rely on a backup when you need it most, it is not a good backup. I can only guess the problem has something to do with her jumping from OS 10.8.5 to 12 Sierra? But it seems to me that what we are trying to do is not extraordinary and a typical situation for which these apps were designed. A big fail for Apple on this one.
This is not the first time recently that I've found a lack of backward compatibility with Apple. Recently my iPhone died and an SE would not sync with my 2007 MacBook Pro. So I switched to Android, which, ironically, does sync. Now my trusty MacBook Pro has finally died and I don't think I'll be replacing it with an Apple product. I've been an Apple user since 1992 with an LC475, but a certain arrogance has crept into the offices of the company that once was close to bankruptcy in the late 1990's. Yes, they have become very successful, and many people are quick to upgrade to the latest phone or device, but losing sight of founding principles such as ease-of-use and reliability, not to mention customer loyalty, has driven this user to look for an alternative.