Honestly? By Bethesda not thinking it through when they came up with the idea of a town of only children. Which is hard for me, because I love fabricating theories that can rationalize any deviancy from the lore.

But for the sake of theory-crafting, let's try and construct a plausible scenario.

First off, they'd have to breed like rabbits. There wont be new kids arriving from outside, so a lack of genetic diversity would doom their community within a few generations if they didn't get to reproducing as soon as possible. Especially when they have a hard-deadline on who can stay in the town. So that means the moment the kids hit puberty they were going at it. This means the older kids in Lamplight were the parents to the younger kids, there's a troubling thought for you.

They'd have to perfect the art of being incognito. A town of children, combat trained or not, wouldn't stand a chance against raiders, much less slavers..or even worse - supermutants. They would have to instill a sense of caution and stealth into each and every member of their group. And accept the fact that any kids caught on the outside would have to be chalked up as a necessary loss to protect the community, too risky to try and save them. A pretty heavy philosophy for a bunch of children.

They'd have to have a very martial system in place for older children who turn 16. Out they go, whether they like it or not. Big Town would serve as a decent incentive to leave (lie or not), but some would still be reluctant to go, regardless. Gotta be able to bring down the hammer on those who challenge they system. Some Lord of the Flies shit going down in Lamplight.

tl;dr: Lamplight is as bad as Paradise Falls.