Crossover distortion at low volumes is very obnoxious because it dominates the signal, but at high volumes it just becomes a harmonic component as it's level will be pretty low relative to the rest of the waveform.
This.
With serious "underbias" distinct crossover distortion would be constantly present in the signal and its magnitude would even be comparatively higher at low signal levels than at higher ones. It would sound like someone "layered" the signal with "buzz/fizz" and the effect would be more obnoxious the lower the volume.
But that's not what's happening there...
What's depicted in the pictures is a different mechanism of crossover distortion: When power tubes get sufficiently overdriven their current draw begins to turn very "asymmetric" and this shifts their DC offset levels. Since DC offset is equal to bias voltage the bias shifts "colder". Now crossover distortion begins to gradually appear when power tubes get highly overdriven AND for a moderate period of time.
So you have basically a time invariant mechanism that adds even more harmonics to power amp's clipping distortion, and the mechanism is also highly interactive with overall envelope of the signal. So it's a very "touch sensitive" process that adds a whole new harmonic pattern to the existing distortion. (Basically one can thus change the "timbre" of the clipping distortion by varying his playing dynamics). Since such process is merely adding more distortion to distortion the effect is indeed very different than that of crossover distortion which is constantly present and adds distortion to clean signals.
Also, whenever you push a tube power amp to such state you have other stuff happening as well: The screen grid current draw of pentode and tetrode output tubes suddenly ramps up (as the tube goes to griod conduction) and this will often modulate the screen voltage of the tube, which in turn results to gain compression within the tube. Plate voltage will also drop so the clipping threshold reduces even further.
Also, whenever one tube "deep saturates" in a push-pull amp the other will "cut off". Cutting off the current flow in a circuit that is transformer coupled to reactive load will create transient ringing, which then add high voltage "spikes" to corners of the clipped wave, corners of the crossover clipping, and corners of wave clipped by OT saturation. Basically to everywhere where current flow suddenly halts. In poorly designed systems these high voltage surges may even arc through and damage the amplifier itself.
SVT, on the other hand, is an interesting case because its output stage design tries to minimize many of these effects: The screen voltage is somewhat regulated to prevent screen current from modulating it, and the power tubes are driven by DC coupled low source impedance high current amplifiers (cathode followers) that actually handle driving power tube grids to conduction rather nicely. (For same reason there is not much of the aforementioned bias shift during excessive overdrive). Additionally the input has diode clipper to "clamp" input voltages below certain level so that it becomes somewhat impossible to drive stages of the power amp to "deep" saturation.
SVT power amp is a very good example of VERY different design goals and practices than what you usually encounter in generic tube guitar/bass amps.