I've taken melatonin a few times-- it put me to sleep, though I think the onset of sleepiness was more like an hour after taking it.

Sure, that's possible. Maybe your formulation simply took a long time; if I chew my pills, they act that much faster since they have that much more surface-area. No surprise there. I've seen time-delayed version as well.

I've heard that people's natural production of melatonin declines with age, so there may be an age below which the pills don't make a huge difference for most people.

This too is possible. I was reading some of the child studies, and an infant or young child produces impressive amounts; supplementing wouldn't do much. (I also tried some on one; he reported sleeping a little bit better but nothing he would've noticed if I hadn't asked him.)