All I ever hear are horror stories about any of these so called roadside assistance plans. I've never had one, never will. I would rather call a local tow company or repair service myself if I ever needed one. That eliminates all the problems with going thru multiple people and agencies.
I would guess that these big companies like AAA have contracts with tow operators for set amounts, thus not a very high profit potential for the tow company. The tow company probably give them rather low priority. I'd rather explain my situation to the tow company directly, then we can together decide on a cost vs. importance scale for my situation.
As for changing a tire, what you need is a cordless impact wrench. I carry a Milwaukee M18 1/2" drive wrench that will bust loose any lug nuts. I change my own tires. The last time I had to I think it took all of about 20 minutes and I was on my way again. There's no way I'd sit around with my family on the side of the road waiting on some company I'd never heard of to maybe show up.