Time Machine Won't Back Up After System Restore After Logic Board Replacement

Check Time Machine settings "Options" and make sure that your drive is not on the exclusion list.

Next, I would run Disk Utility->First Aid on the drive to see if there is any problem and if it can correct it. If it finds errors and cannot correct them, then the drive could possibly be damaged. In that case, you could try and see if you can successfully erase the drive and then try a new Time Machine backup. However, I personally would not trust it as it could still be an accident waiting to happen because of your power surge issue.

I don't know if you went through this iteration but try removing the drive from Time Machine settings, property eject the drive. Wait several minutes then plug the drive back in. If not immediately recognized as a Time Machine drive re-add it to Time Machine settings.

Sometimes continuing Time Machine backups when getting a new machine (replacing the logic board essentially is giving you a new machine) can be problematic and ultimately the only solution is to erase the drive start anew.

I’d guess the Passport took a hit too but problems didn’t crop up right away. Eliminate that theory? Use another disk to verify.