Originally Posted by LesterOfPuppets

Turns out it was the headset all along. It was strange in that if your headset is loose enough to rattle, you usually just need to barely rock the bike back and forth with the front brake applied to see the headset parts move, but it didn't move unless I really rocked it good. The cup was really resistant to adjustment also, like gunked up threads (or just 35-year-old parts that haven't moved since initial assembly), but once I got it moving I tightened it a little over a turn 'til it was adjusted properly.

I also repacked the front hub while I was at it, and got rid of 22.8 grams of dead weight; drastic average speed improvement is expected: