well, AAA is known as the premier choice for roadside assistance. Every time i've needed a tow, I ask the tow drivers what's the best premier choice that they'll jump to get calls for and get paid and they always say AAA. But AAA is also regional so AAA in one region might not be exactly the same as AAA in another.
So for the OP's particular incident, I don't believe any other roadside assistance choice would've been able to get you any better result.
There may have been some communications hedging. Often times, the dispatchers for YOUR service, always greatly hedge their estimates, so they don't disappoint you. The tow truck company says 1hr even though they can be there in 30, then dispatcher relays to you 1.5hrs because they don't trust the tow company. So potentially the AAA would've been the same 30min call as what the cop got for you.