After searching the internet for days I did finally try what I thought was an unlikely cause to this problem -- I changed out the video card. This solved the problem.
My specific problem was that when I pressed the power button the light was lit (no red diagnostic lights on the logic board) but after a brief spinning sound there was no startup chime and there was silence until the fans started up full blast. I had been using the computer and it suddenly turned itself off and I was never able to restart it until I replaced the video card.
If you want to test to see if this might be your problem -- before you spend money on a video card -- take the video card out of your machine and press the startup button. If you get the normal powering up sounds with normal fan noise it is likely the video card.
I had read this solution in one or more posts to this issue, but I couldn't see how that would work so passed over trying it in favor of other possible causes. I am posting this so others will take note that this is good advice even if it seems unlikely.