They said that AAA won't cover the tow because because it didn't have valid plates or insurance. Now you claim it had valid plates but don't seem to deny that it didn't have insurance.
So the question is if there is no insurance,which in most states is illegal to drive a vehicle with no insurance, how did it end up in a ditch?
If this car was in a ditch, it does not sound like it was blocking traffic. If that is the case why did the State Police need to stay?
Then after all of this and you needed to tow it home, why didn't you call AAA at that point instead of paying $50? If you have proof of registration and it really was insured it does not matter what he tells them, because YOU have the proof. Because if MI is anything like CA you need to have a copy of your registration and insurance with the vehicle. So it wouldn't be like you "left it at home".