The idea that fuses and power supply components are not part of the signal chain is not entirely correct. Anything coming in from the A/C, power cables, fuses, rectifiers, regulators, or power supplies can all become part of the output signal.
An operational amplifier circuit (whether discrete or monolithic op amp) will only pass through the actual signal if the signal is coming in on the inverting input of the op amp AND the negative feedback gain is unity gain (0 gain) or less than unity gain (i.e. it’s forcing op amp to de-amplify the signal). In this case, the waveform signal will actually route itself around the op amp using the negative feedback line as a pathway. That being said, even with unity gain, the type of op amp or power supply elements will still influence the character or sonic signature of the audio.
If the negative feedback circuit is set to create a "gain" (or amplify), it increases the voltage of the waveform by using voltage from the positive/negative rails from the power supply (i.e. +/-15V). This additional voltage is sent through the output pin of the op amp. It this case, the fuse, power cord, A/C elements will contribute to the character of the waveform (in the sense of affecting how the op amp slews in relating to the input waveform, how far/fast it slews, character of the DC noise, etc.).
In a power amp circuit, the output transistors have to create massive gain (turning a 1V input into something like 15-100 watts or more). It has to use the A/C power coming in to create this voltage, so your signal is actually something like 98% A/C voltage.
https://www.youtube.com/watch?v=7FYHt5XviKc
I’ve posted that link before, but it’s an excellent lecture on how op amps actually work.