Migration Assistant Problem - External Hard Drive

I have a Mac that's almost ten years old but it has worked great until recently. It started having problems in Monterey. I used Time Machine to create a backup then started trying to diagnose the problem. This led me to erasing its Macintosh HD and attempting a re-install of Monterey.

Then it wouldn't accept my re-install so I lugged the thing over to the Genius Bar. A bright young guy there managed to get an earlier operating system installed then advised me to upgrade to Catalina then use Migration Assistant to complete the recovery.

Migration Assistant gets stuck trying to find the drive even though it's the only USB drive plugged into the computer. The wheels just spin and spin.

I have my USB drive plugged in and I can see it on the desktop, all decked out in Time Machine green. I can click Get Info and see that it is holding the right amount of information. However, when I open it, it appears empty.

If I go to Recovery Mode and attempt to restore from there, I can get to the drive, but it appears empty there as well.

The only thing I can guess is that Catalina can't see what Monterey made. I'm hoping someone can reassure me I didn't just lose everything and offer a solution. My only glimmer of hope is that I can see that the Time Machine external drive is holding...something I could see in Monterey but can't see in Catalina.