Put it all back together this evening. Sorry, I didn't take photos. I cleaned up everything with acetone – outside, on this fine 42F day – after scraping the original grease out with toothpicks. It was like tar! And had accumulated between the jaws, just where it would be no use.
Minimal wear on the jaw and split nut threads. The parts of the jaws that contact the drill bits were as good as new. Lubed everything and now it operates like butter! It wasn't even this good when new. I'll have to do the drill press chuck now. And maybe all my hand drill chucks. I definitely won't wait another 15 years before the next service.
(Yeah, my thinking above on the split nut was silly.)