Migration Assistant endlessly "Looking for other computers" after connecting to Time Machine

The proper way to restore files backed up by Time Machine is to use the Time Machine app in the Applications folder. Migration Assistant is not a backup and restore utility. See How to Use Migration Assistant with macOS Catalina and How to use Migration Assistant to transfer files from another Mac.

See the following about using Time Machine: Back up your Mac with Time Machine and Restore items backed up with Time Machine on Mac - Apple Support.

The first two are specifically about using MA. But MA is not a general backup/restore tool. That is what Time Machine and the Time Machine application are intended. This does not mean that you cannot use MA with a Time Machine backup, however, MA is generally used when you need to transfer user accounts and/or third-party applications from another computer. MA can perform a full system transfer from a Time Machine full system backup when restoring the backup to another computer or another partition on the source computer.

You haven't provided information on your specific need so as to inform you of what you should do. Hence, I provided links for both Time Machine and MA.

The default should be SMB, now. AFP has been deprecated. I think it is still there for older machines that cannot work well with SMB or don't work at all.

Be sure to read through the article on using MA because you can select two options for the connection but one is confusing and doesn't work. I suspect that may be your problem so you may want to try the other option to see if you have better luck. Also, MA should be run on the Target, not the Source, machine.