Thanks for the suggestions, guys. The good news is that I've gotten
the card to work in two other motherboards now, one of which the card
DIDN'T work on before, which indicates that the card is not itself
entirely broken. But it still continues to light the LED when it's in
my own shuttle AGP slot. I'm still totally puzzled as to why this is.
As many of you suggested, seating is commonly an issue with AGP cards.
I've had problems with AGP cards not seating properly before, so I
paid special attention to that, and I've made a visual inspection of
the contacts along the bottom and on the PCB itself.

What I'm confused on is why it would be broken enough that it wrongly
sets off the 3.3V AGP LED but can still work on 2 out of 3
motherboards. I'll keep looking for any obviously physical defects on
the card.