That video was an extreme example to demonstrate how tube amplifiers overdrive. The beauty of a tube power section lies in what Pat was calling the step that forms at the zero crossing as the input signal increases in size. That step is known as a crossover "notch." That notch produces crossover distortion. It is the result of the amp shifting from class AB operation to class B operation. Push-pull tube audio amplifiers are biased such that they operate in class A, class AB, or class B. In class A operation, both halves of the push-pull circuit conduct current for 360 degrees (i.e., they are always on). In class AB operation, each tube conducts current for more than 180 degrees, but less than 360 degrees. In class B operation, each tube conducts current for 180 degrees. The important thing to remember is that the halves of the push-pull circuit are 180 degrees out of phase. This phase difference is made possible by a circuit known as a phase inverter, which produces a signal in phase with the input signal and a signal 180 degrees out of phase with the input signal. By operating in class B, each half of the power section only has to amplify one-half of the signal, that is, each half of the power section is on only on for half of the time, which results in significantly more power being sent to the speaker instead of being wasted as heat.
As an aside, a post-phase inverter master volume works by progressively summing the two output phases until the output is zero (or darn near it). Two signals of the same amplitude that are 180 degrees out of phase with each other sum to zero.
Anyway, the magic of an amp that can shift from class AB to class B under heavy load is that allows a player to punch through the mix when bearing down hard on the guitar during a solo because an output section is more efficient in class B than it is in class AB (most long-time tube amp players complain about not being able to punch through the mix with a solid-state amp because most solid-state amps remain in their operating class under heavy load). When pushed harder into class B, the amp draws more power than the power supply can provide, which results in the power supply sagging, lowering the plate voltage, resulting in sustain while the power supply recovers. This phenomenon is responsible for the fluidity of a cranked tube amp when playing single notes.
As far as the dissonance problem with non-perfect intervals, that is the result of intermodulation distortion, which more common in high-gain solid-state circuits than tube power amps. Intermodulation distortion which is the result of harmonics mixing, w produces sum and difference frequencies that are not multiples of the original frequency. Intermodulation distortion is very common in two-guitar bands where both guitarists play with a highly distorted guitar signal. Intermodulation distortion is responsible for muddy sounding mixes.
By the way, I challenge you to say that "Cliffs of Dover" lacks clarity. It was recorded using an Echoplex, Butler Tube Driver, and a cranked Marshall.
A lot of the harshness in the Pat Quilter video is due to the JCM 800 being voiced much brighter than previous Marshalls. The amp needed to be in the era of synth-heavy music that was the 80s.
One last thing, while I still own one of the tube amps that I designed and built in the mid-90s, I am not a tube snob. I own a Tech 21 Trademark 60 and a Quilter MicroPro Mach 2. I appreciate both technologies, but tube amps own playing dynamics. I really wanted to like the Blues Cube because it is allegedly modeled after the 5F6-A Tweed Bassman, which is the amp circuit from which almost all Marshall amps were derived. While the Artist does not sound bad, it does not remotely sound or feel like a Tweed Bassman, regardless of the hype. Joe Bonamassa is playing straight into two high-power Tweed Twins and two Tweed Bassmen in this video: