Time Capsule problem

Don't get rid of your old Mac as it can be the way out of the problem.

It should be possible to plug the old Mac into the new Mac although you will need to buy a special cable.

Perhaps the easiest way is to create a clone of your old Mac on a USB drive. So you will need a disk that is same size or larger than the one in the Mac. Make sure it has suitable USB fitting to plug into the new Mac. You can use Time Machine even.. but I prefer a clone software like Carbon Copy Cloner. It is free to use for a month and I do recommend it in place of Time Machine now.

Once you create the USB drive backup or clone.. plug the drive into the new MBA and try again with Migration Assistant.

Local disks like ones plugged into USB are treated completely differently to network drives.

Migration Assistant is not going to work unless your "new" Mac can "see" the Time Capsule on your network. There are several ways to check on this as follows:

Method 1

Open AirPort Utility on your Mac

Does the Time Capsule appear? If yes, click on it

Click Edit in the smaller window that appears

Click the Disks tab at the top of the next window

The Time Capsule disk......named "Data"......unless you have changed the name of the drive will appear

Check using Migration Assistant to see if it can now find the Time Capsule drive

Method 2

Open any Finder window on your "new" Mac

Look under the Locations heading on the left side panel to find the name of your Time Capsule

Click on the Time Capsule and another window will appear to the right to display the "Data" drive on the Time Capsule

Double-Click on "Data" to mount the Time Capsule drive on your desktop. You may need to click Connect As and enter the administrator password for the Time Capsule

Check using Migration Assistant to see if it can now find the Time Capsule drive

You don't indicate what operating system the "new" Mac is using, but we should report that there are a number of users who are having problems with Catalina and Time Machine Backups, so that may be another separate issue here if your Mac is running Catalina.

Apple changed the format on the Mac hard drive with Catalina to APFS. Previous Macs used HFS+, also known as Mac OS Extended (Journaled). The Time Capsule used the same HFS+ format.

The issue appears to be that Catalina has varying degrees of success when it tries to read older Time Machine backups that have made from an HFS+ Mac to the HFS+ drive on the Time Capsule.

I don't have a "fix" for you on that issue, but another user may be able to help.

Not sure if Apple Support is available at this time, but you do have 90 days of free phone support with the new Mac, so if you have not already done so, a call to Apple Support might provide some other things to try. Good luck!

Hopefully, you are not using Catalina on the Mac. Can you provide more details?