I dont have experience with AAA and my tesla for flat tires but (unfortunately) have had 3 separate occasions to use Tesla roadside assistance for Flat tires / bent rims on my Model 3 Performance. I called (once) and used the app (the other 2 times) Tesla for those as my car is still under warranty.
In all three cases, the towtruck that arrived was from the "prominant" tow comapny in the area I was in, and it displayed (in large font) "AAA" logos on the towtruck. Meaning, had I called AAA, the same company would have sent the same towtruck, most likely.
In two of the three cases they had a loaner wheel, and in the third case I was actually close enough to home that I had my wife drive over with a spare wheel I had in my garage (because I have zero g referral wheels on my model 3, and still have the factory wheels / tires in my garage).