Light,
For an "amateur", you seem to get it pretty well
.
The transformer uses a source of DC voltage to source current. The tubes can't pass it, as tubes can only pass AC. In a single-ended amp, one end of the transformer is free, and can be connected directly to the power supply. In a push-pull amp, both ends have tubes on them, so a connection to the power supply is made in the electrical center of the primary winding, allowing the transformer to source the current it needs.
Sag is exactly how Light described it, and is a complex relationship between the rectifier, output transformer, and filter capacitors. The rectifier has the biggest effect, in an otherwise properly (in my mind) designed amp. A tube rectifier contributes to sag. A solid-state one does not.
Many players love an amp with a certain amount of sag, for the compression and sustain Light described. Some describe it as making the amp "sing". Hit a chord or note hard, there is an initial transient, followed by gain reduction (sag) and then sustain as the amp comes back to full power. These people feel solid-state rectifiers sound too "hard". Some prefer a solid-state rectifier or even a fully solid-state amp, because with high-speed playing, an amp with a lot of sag won't have good articulation. The amp can't deliver full power to each note played.
This is the reason for things like the Mesa/Boogie dual and triple rectifier amps. Multiple tube rectifiers keep the sag minimized at loud volumes and fast playing, but there is still some of the tube rectifier's characteristic sing and sustain.
As far as harmonics, I will assume you know what they are. Electronics produce harmonics above (and sometimes below) the fundamental of any signal that passes through them. Just the way it is, no getting around it. These harmonics are considered to be distortion. Some harmonic distortion is good in a guitar amp.
A word about distortion in this context- any deviation between the input and output signal of a device, like an amp, or mic pre, or mixer, is called distortion. The waveform is distorted, or changed. It doesn't mean overdrive or pedal type distortion. The most common source of distortion is harmonics induced in the circuit electronics. It is so common that most gear has a %THD, or Percent Total Harmonic Distortion spec. There is another kind, called intermodulation distortion, which relates how two signals interact in a device, but here the biggie is harmonic distortion.
Much is made of the difference between even and odd-order harmonic distortion. Though not 100% correct, and very simplified, the basic idea is that even sounds good, odd sounds bad, and that tube circuits make even-order harmonics, and solid-state makes odd. Like I said, not even really close to correct, but sort of a basic place to start to get the idea.
A certain amount of even-order harmonic distortion just plain sounds good in a guitar amp. It is ear-pleasing, giving a rich sound. The guitar amp is part of the sound, adding it's own character. In a hi-fi amp, the goal is as little distortion as possible, for the most accurate reproduction of the music, while the amp remains transparent. A guitar amp with no harmonic distortion sounds cold and sterile. Think of a really bad solid-state guitar amp, and how much emphasis is placed on how much a given solid-state amp "sounds like tube amp". This is done by trying to get rid of odd-order harmonic distortion, while adding even-order, in part. It must be pretty hard, as no one has done it really well yet, though some of the new SS stuff by people like Vox sounds good enough not to make me puke.
A push-pull amp by it's nature cancels out some harmonic distortion, both even and odd, and with other design factors, you can get the distortion pretty low, or add in 2nd order distortion. And also by nature has less %THD than a single-ended design, and so less of the pleasing 2nd order distortion. That is what is behind the statement that guy made. Thankfully, guitar amp makers don't go beyond the natural characteristics of the push-pull design, at least in tube amps.
These days, anyway. Rumor has it Leo Fender was "horrified" that people were turning his amps up far enough to distort, as one of his goals was to keep distortion to a minimum, as he favored country players, and designed many of his early amps for them. Hearing Dick Dale and Link Wray must have just about killed him, as he strove for a couple of years to completely eliminate distortion in his amps, before caving to the inevitable fact that people liked it.