Does it matter which component I power up first?

FWIW, I do precisely the opposite of what is being prescribed here. I turn my amp on FIRST, before the other components, and I turn it off LAST.

It's an integrated amp (Decware Zen Torii). I keep the volume adjustment knob turned all the way down during power on & off, and I've never heard any audible "pop" from the speakers. Not when powering on/off the amp, other components, or when switching sources.

As I have no audible pop, I see no reason to alter my process.

Preamp, sources, power amp last on. Power amp off first. Don't believe the rest of us? Do it your way with the volume all the way up.

My owners manual says to turn on the sources on first, then the amp. To power down: amp first, then sources.

Follow you owner's manual advice. You can probably avoid the turn-on thump problems doing as Lupinthe3rd does and muting or turning the volume all the way down on the integrated amp, but you do risk a disturbing sound if you forget to do that, and some mute circuits don't entirely cut off signals, depending on the design.

It isn't that I don't "believe" you. With the volume turned all the way down, the claimed issue is nonexistent. Why would I want to do it with volume all the way up?