Well, it took two years and three months, but I finally became a victim of the "Red Ring of Death:"
So I’ve contacted Microsoft support and they (hopefully) will repair it. I can’t make it "red light on demand" yet. But pretty much, if I play for just a few minutes it stops working.
This seems to be happing pretty frequently lately, as I’ve gotten a few messages from friends saying "Sorry, I’ll be missing for a few weeks while my system is repaired." So it looks like all the original systems must have been affected, it’s just taking its good old time on some boxes.