Hah. I (many, many, years ago) had the literal opposite of this happen, also thanks to phoning the AA. I was on the M1, car temp shot up, so I pulled over, waited 4-5h for the AA who never showed but they sent a local garage tow truck. Driver peered under bonnet, said it's probably the head gasket love, don't drive it (with no checking anything else) and then charged me £150 (cash on spot) to tow me to my folks. That was terrifying as I didn't know this guy and I had to sit in the truck for 50 miles with him.
I called the nearest main dealer garage the following day. They turned up, peered under the bonnet, removed the roll of masking tape from the fan that the service folk back home had left there 2 days previously, laughed and then drove it off for checks. Cost me a fortune that as a postgrad student I didn't have. Could barely afford the car.
I wrote a very stiff letter to the AA noting that if their so-called partner engineer had checked properly, I'd not be out £150 and that I expected better etc. and I wanted my money back. They refused to take any liability and offered me a free year's membership, which was at the time £38. I think I may have told them to stuff it.
I've been with the RAC ever since.
Good luck with your complaint, but I wouldn't hold your breath.