IMHO just start over.. for sure.. keep the old disk as archive for a while.. but copying backups from local to network drive adds complexity. TM does not backup to network drive exactly the same as to local drive.
USB drive copied via air to TC is certainly possible.. In the TC give a different name to 5ghz wireless so you can force the air to use it.. 5ghz is substantially faster as long as you are in the same room with it.
Or buy the thunderbolt to ethernet cable.. and use that.. connect the external drive to your newer air.. that should be very fast. Watch out for thunderbolt issues though.. it is not always smooth sailing.
Read Q18 as referenced above.. it is most important that you create the correct sparsebundle to store the local backup.