Migration assistant is unable to connect to Time machine backup on a shared drive.

I have a MacBook Pro 2011 which is no longer starting up (Motherboard issue). It does start in Target disk mode. I do have a Time machine (TM) back up of it on a USB 2 drive.

I borrowed a friend's old MB pro (also 2011, MacOS 10.13.6) and have connected the Time machine drive to it and shared both the drive and the latest backup folder.

My new computer is a MB pro as well with MacOS 11.2.3. The new computer is able to connect to my friend's shared folders and drive over WiFi and see the TM drive and folders.

I'm trying to use Migration assistant on the new computer to find the shared TM drive. It does not show up automatically. I can connect to it using either afp or smb but I'm still not seeing the backups.

If I start Migration assistant on both computers the new one can connect but my data is not on my friend's computer so it's of no use. The data is on the attached disk.

I've thought about using Target disk mode but the old computer has FireWire 1 and 2 and USB 2. The new one has USB 3 connector only. I'd rather not buy a cable for this only use.

What might I have do, to get the new computer to find the shared old disk with the TM backup?

Thank you.