I haven't had a flat in decades. I think the last one I had was the late 80's - had a '79 Monza at the time. Wife, on the other hand has had a few. Was worth it to have AAA or a motor club membership. At least she could call and they would change the tire. One time she was fortunate to have the Road Safety Patrol right behind her when a tire went down.
I had AAA for quite some time. Rarely had to use it for "roadside assistance". Used It for other discounts and maps, TripTiks, etc. AAA, like other motor clubs, is not perfect and yes, in times of severe weather, they may not come out or be delayed. Happened to me 20+ years ago. Living in Denver at the time we got a severe cold spell (subzero temps for over a week at a time - highs were -15*F & lows -35*F). Due to the high demand for jump starts and other assistance, AAA was overwhelmed and advised (if you called) it would be 72-96 hours before they could get someone to you and you'd probably get faster service by contacting a local service station or garage.
We have "Towing" on our insurance, but it only pays $50 per occurrence. Considering that's what a tow driver charges just for showing up, it ain't enough. At least with AAA+, you can get a tow up to 100 miles.