In the context of this circuit:
Class B is subject to much misunderstanding. It is often said that a pair of output transistors operated without any bias are working in Class-B, and therefore generate severe crossover distortion. In fact, with no bias each output device is operating for slightly less than half the time, and the question arises as to whether it would not be more accurate to call this Class-C and reserve Class-B for that condition of quiescent current which eliminates, or rather minimises, the crossover artefacts.
In other words, given that the crossover distortion is gross enough to be seen on a 'scope, that absolutely makes it class C - no ifs, no buts. If we were debating something that needed a distortion or spectrum analyser to distinguish, then yes, perhaps we might be in the pedantic zone...

As you suggest, class AB is about avoiding the lesser of two evils (gm-doubling (AB) vs crossover distortion (C)). Pragmatic enough, especially if the rest of the design isn't all that clean...
In a better amplifier, maintaining that exact bias point is worth doing, but is one of the harder things to get right. So much so that an entire chapter is devoted to it (chapter 13). For an EF stage, the quiescent voltage needs to be held to around 100mV, and for a CPF, it's 10mV (ballpark figures) - obviously very easy if these were simple fixed voltages, but of course these voltages must track the changes in output stage Vbe caused by temperature. It can be done, and Douglas Self has put many such designs into production - and I'm sure he's not the only one.