Migration Assistant and USB Time Machine disk Mavericks Update

Sadly this issue remains unresolved in Mavericks.

End Update

When I run Migration Assistant (from an admin account) in Lion, M.A. hangs at the message

"Looking for Other Computers". The Time Machine disk is not seen even though it is connected through USB and it is the Lion TM disk from the same machine.

I tried:


Pressing the greyed-out 'continue' button.
Having the Time Machine both selected as the TM disk, or connected
without being selected.
Rebooting into the recovery partition and attempting recovery from
TM. At least in that case the TM disk is seen, but it cannot be used
because "only full Mac OS X backups can be restored". I only save
user files on TM.


What are some other ideas I could try? I prefer to avoid running a 'cp -pR', but if you think I could get away with it, please mention the caveats.
I've had a similar problem, which may be different to your problem because all the computers in my house share the same backup drive, and so there are lots of different backup files. 

My problem (which I fixed) was that Migration Assistant couldn't see the backup drive, but it turned out that if I opened the drive in Finder, and then opened the '.sparsebundle' file that belonged to my computer, then all of a sudden the backup drive became visible to Migration Assistant and I could restore from backup.