Well, the source of the intial failure(s) is TWO bad video cards! I took the good-looking card to a local busines named BCOM. Since I was the one and only customer in the place, I walked right up to the computer technician and repeated all of what I've posted above -- albeit in a very abbreviated, right to the point fashion. At the end of the description, I said, "is it possible to test to see if the fan works?" He did not say one word. He just took the card out of my hand, quickly removed the short metal plate at the connector end of the card and walked over to a system that was just sitting there, case open, probably a testing unit for maybe just this kind of test. He quickly inserted the card and pushed the system's power butting. He and I heard some beeps, the fan did not start, and within 10 or 15 seconds he declared the the video card was bad. I asked him if it was likely that the card had damaged the PCI-e slot on the computer and he said "probably not".
That was enough for me. I walked over to the salesman who had been listening to this test and he was already looking for a replacement card. Believe it or not, he recommended one that lespurgeon had proposed above --A GeForce 210 Silent:
Bcom wanted $40 for it, so because I had been treated so well, I instantly bought it.
To make an already-too-long story short, the system is now running in our home theatre downstairs, sitting right where it always has.
I must say that I like the 8500 GT's nVidia control panel better than the 210's.
First, we drive two display devices with the HTPC -- the RS1 and an older Viewsonic 1368 x720 LCD monitor. The older control panel had a so-called "clone" mode, where the Viewsonic's and RS1's display resolution were matched and the Vista desktop icons apperard on both displays. As far as I can tell so far, this mode is not available on the new control panel, so the display that is set to "primary" is the display that displays the icons. But the desktop background is indeed displayed on the secondary monitor, so maybe I've missed something. The problem with this is that after watching a movie on the RS1 and changing it back to the seconday device, I then have to go into the other room, turn on the Viewsonic, go back to the RS1, change it to secondary and then rush back to the Viewsonic to "accept" the changes. This is not a huge deal, but it is a PIA.
Second, there apparently is not 24 Hz setting available with this card. 25 Hz is the lowest available, so that's what I selected. I was very concerned about this, but after I started watching a movie on the RS1, after a bit of time for ReClock to do its thing, the movie smoothed out to a point where neither my wife nor I could see any defect whatsoever in the smoothness of the video. The image quality is at least as good as the 8500 GT. When it gets this good, it's hard to discern which is the better card.
After downloading the latest drivers for the card, I did a brief search for the specifications of the 210 and I couldn't find any that listed the available frequencies of this card. Too bad.
Because the 210 cost so little, I'd consider buying the 430, or any other nVidia card, if I coiuld determine before buying whether it supported a frequency of 24 or 23.976 Hz. Maybe someone reading this knows a contemporary, not too expensive, nVidia card that can do that. If so, please let me know.
Anyway, gentlemen, and possibly ladies, I thank all of you for your help. The problem turned out to be a simple video card failure. But I never would have imagined that the main symptom of such a failure might be a system's failure to do absolutely anything except give a little flash of yellow light and the bump of a cpu fan when a power cord was plugged in.
An uneccessary impediment to troubleshooting was HP covering up that on-board VGA port with that ridiculous plastic cover.
So to anyone who has a Slimline of similar vintage that is behaving in a simlar manner, step 1 is remove the video card, plug in a monitor to that hidden vga port and plug it in again.
PS, Now I'm going to have to tell my friend what his system's problem likely was/is and ask him if he'd like to bring his dead system back to life. (To say that his system is "disassembled" is putting it mildly.)