Two problems with this argument-
AAA capable of shooting that high is large and prominent, like that 88mm L71. And they require a lot of ammo, hundreds of rounds to get a hit. That was a big gun, and a big target, and a big logistics footprint.
An AAA battery will require a couple of radars to find the target and point the guns, this can’t be done by eye, even in WWII the Germans were using complex integrated radar systems for this. Radar also shouts “shoot here”.