If this were my neighbor, I'd advise her to kick him out the door and call a firm across town. Scammers don't deserve any money and checks for service call fees can be stopped.

Stupid question, though. The furnaces I've seen have had transformers to power the board. Was the repairman sure that the transformer wasn't going bad?

BTW- I wouldn't blame the repairman. He's probably salaried and doesn't give a hoot. I'd blame management for pushing him to make more sales -- there's something wrong higher than him.