That is so true. Once I determined that the receiver had a repeat of the service need that it had a few years ago, described by someone here as probably a solder reflow, the client wanted to throw it out the window. I had no need to pursue information any further. I have only continued to answer questions here for the benefit of anyone else working with the same pile of crap.
You write "an AVR" as though you've done research and determined that THIS AVR behaves that way. According to the manual, you have to connect to the main HDMI output to see the setup image. It might be true that "an analog" video output had video when in setup mode... but since we were already going to trash the unit, why would I research whether the NTSC, S, or Component outputs had video?

Actually I've been wondering when someone was going to point out to me that
I missed that, per the manual, when there's no HDMI output, you can use the display on the front of the receiver to track where you are in the setup process.

But after I had reset the receiver and still had no audio output, I'd say "since I got the receiver reset to factory default and the power amps were still dead, why would I spend any more time trying to set it up?

"Whose" has two definitions. You cite the first one. The second one is more general:
This definition, in writing "noun," includes things that are not persons.

There's another subtlety here. You are absolutely right about "its power supply blew out." But if you want to write a sentence phrased as I did,
"...his HDFlow... whose power supply blew out...."
How can you write that with "its" ? That is, would one* write
"...his HDFlow... XXX power supply blew out...."

What word goes in that XXX spot? Its? Their?

Another thing. There's a difficulty since the plural of "it" is "they."
Yes, "its power supply blew out,"
which clearly refers to a thing,
but
"their power supplies blew out."

That use of a personal pronoun is clearly a reference to a thing.
We use "they" to refer to people AND things, so why not accept that we sometimes use "whose" to refer to things?

*let's not go there -- one has covered that already.