The power supply should assert a power good signal when the supply lines are stable
True. The "PowerOK" signal should go to a "logic HIGH" (3.5 to 5.0 volts) within 500 milliseconds from powerup to meet ATX 2.0 specs. This removes a hardware RESET from the CPU enabling it to start executing BIOS code.
but it would appear that the supply gives a glitch (maybe caused by the hard drive starting up) causing the computer to crash.
The "PowerOK signal drops to a Logic ZERO forcing a hardware RESET.
Possible causes of this could be a faulty or inadequately rated power supply, Very likely.
running the computer from a long extension cable,
Unlikely. My work computer, a medium grade C2D system, is powered through a 25 foot extension cable, and two outlet strips with 6 foot cords. (My shop office has power distribution problems.) A good PSU will power a computer with as low as 90 volts going into it. You would need a really long extension cord.
faulty motherboard, Possible. And it's possible that the onboard video circuits are glitchy. If that's the case, if the motherboard has a slot for a video card, you can use an external video card.
or faulty processor. Unlikely. First, CPU's are very reliable. Second, a bad CPU would tend to hang in the same place every time.
Now for troubleshooting, you can try this (the following is a general troubleshooting guide I wrote, not everything will apply to your problem):
Clear the CMOS RAM. (This works often enough to be worth doing.) Your motherboard manual will tell you how. Make sure the HSF is properly installed. (You may have to remove the motherboard from the case to check this.) Turn on your PC. If it's still broke, continue on.
Disconnect and unplug everything but the PSU, CPU & HSF, power & reset switches, and the system speaker from the motherboard. When you turn on the PC, you should hear a series of long single beeps. This will indicate a memory failure. (You do not have any memory installed, remember?) In a way, this is very good. This tells you that your PSU and CPU are good, and the motherboard is probably good.
If you do not see anything happening, swap the case power and reset switches. (Maybe you have a bad case power switch).
If you get silence, either the PSU, motherboard, or CPU is bad. The only practical way to determine which is bad is to test by substitution. Most likely failure (in order) is PSU, motherboard, and CPU. One other possibility is that you have something installed improperly in the case and it's shorting out the PSU. Only way to test this is to remove the motherboard from the case and reassemble everything on an insulated surface. This is called "breadboarding" (from the '20's).
If you hear the beeps, turn off the PC and install one memory module. Turn on the PC and you should hear one long and two or three short beeps indicating a problem with the video card. Silence indicates that the memory module is shorting out the PSU. Long single beeps (my GA-EP35-DS3P and eVGA 680i motherboards) usually indicate a really bad memory module. (Your BIOS codes may be different.) Test and install the rest of the memory.
Turn off the computer. Install the video card and plug in the monitor. Turn on. System should boot and pass POST (single short beep), and you should see messages on the screen. If not, your video card or monitor is bad. A bad video card will usually generate something like one long and two or three short beeps. If you get silence, your video card is probably shorting out the PSU. Again, your BIOS codes may be different.
If you see messages, turn off PC and plug in keyboard and mouse.
If this works, start plugging the other components one by one.
This can all be done before installing any the parts into the case. I always breadboard a new build. It lets me test all the parts before I install them in a case.