> The tonal characteristics of a pickup are inherent in the voltage wave.
Thanks for getting that. I saw your confusion and went to bed instead.
The "tonal characteristics" are hard to understand. However whatEVER they are, if we can make a duplicate signal which is EXACTLY the same only BIGGER, we get nearly the same tonal characteristics only LOUDER.
So the amplifier only has to be "exact".
And it only has to be better than the final use of the large wave. Our guitars don't make output below 82Hz. Our ear-bones don't work much past 20,000 shakes per second; for other reasons, guitar need not go this high. The ear is not a precise judge of amplitude. An amplifier that has reasonably consistent gain from 80Hz to 10KHz, and does not add significant amount of other tones (hiss, hum, buzz, distortion) is "perfect enough".
The tube is stupid. Imagine we custom-build a stupid demon. His left arm has a very delicate finger which can rest on your string and follow its vibration without affecting the string. The right arm is powerful. We train the demon to shake his right arm "exactly" the same as his left finger, only stronger. We tie that big-shake arm to an air-pusher. Small string vibrations are turned into large air vibrations.
(In concept, a person could do this. In practice, a human's fastest arm-shake is slower than a slow string vibration. Our frequency response is poor. Our finger-arm coordination is sloppy too. My imaginary demon could be imagined to be as fast and as precise as needed. Tubes are good for low gain to 1,000,000Hz, high gain to 10,000Hz, so audio vibrations are not hard for them.)
Note that the demon is not doing anything "smart", does not have to understand "tonal characteristics". Just has to output what comes in only stronger.
All this strong work makes the demon hungry. Amplifiers have to be fed. Our demons (tubes and transistors) eat a steady electric current such as from a battery (or similar derived from wall-power). I assume a demon has a liver to store food energy between meals. Tubes don't have livers. On battery power, the battery can supply energy as needed. On wall-power, food only comes out of the wall in pulses every 8/1000th seconds, we need a power capacitor to "liver" the energy between food-pulses.