The cross over is there. You're just not seeing it. Take a look at this (attached):
Here, you see the output of a vacuum tube amp I designed and built subtracted from the input signal with a differential amp. The residual is a sine due to phase shift. Here, you can see the switching glitches caused as one VT in a push-pull pair cuts off. The mode here is Class AB1. If you o'scope the output alone, you don't see these glitches since they are so small relative to the output.
If you were to do the same, subtract the input from the output and null the difference as much as possible, you'd see plenty of cross-over distortion. If you hooked up a small speaker, you could hear it. Cross over sounds really nasty.