Some people might say that stuffing a heating vent full of carpet underlayment is bad for airflow. Some people might. Not the former owners of this house though, no sir, not them.
I think what happened was that the sides of the vent pulled away from the opening (you can see this at the top somewhat) and, rather than fix it properly or even investigate what was happening, they assumed that the draft could be solved by just blocking up the vent entirely. I guess.
Anyway, I pulled out all (I hope) of the underlayment, screwed the vent into the wood properly, and taped things up nice and draft-proof. Should help keep the temperature a bit more even in the living room area now.