'fraid you're just plain wrong about that. While what your illustration shows is indeed crossover distortion, what is depicted is more along the lines of what a Class B design would yield. Class AB is designed to have a little bit of bias current running so that the transistors do not have to turn on, as they are already idling. An optimally biased Class AB amplifier can have vanishingly low distortion.
The real problem is the transistor, which has a gain that changes with the amount of current flowing through it. In class A, this current is fixed, thus less variance in gain (and lower distortion). Modern audio transitors go a long way towards solving this problem, as they are more linear.
Mmm, that's interesting. I'd heard some of that, but did not know that switching was not a problem in A/B amps.