When I first read "The Lottery" I assumed that the children assembled first because they considered it an exciting event and had been looking forward to it. The boys gathered a lot of rocks for the same reason. They thought it was a fun event. It didn't occur to them that one of them could be this year's human sacrifice. They enjoyed stoning someone to death. They really didn't understand what was happening. They didn't understand death. They looked forward to it and talked about it for a long time afterwards. Of course the children would be the ones to gather first. As far as the adults were concerned, they were not looking forward to it. They were in no hurry to get there. They knew the odds against being the chosen victim were great, approximately three hundred to one; but just the same, there was always the chance that they could find themselves in the position of Tessie Hutchinson, with all her relatives, friends, and neighbors suddenly turning against her and closing in from all sides.
The men were "speaking of the planting and rain, tractors and taxes." The women "exchanged bits of gossip." The adults are obviously not excited about this event. The man in charge treats it like a grim necessity.
"Well, now," Mr. Summers said soberly, "guess we better get started, get this over with, so's we can go back to work."
This is a great question. In this story, this town (and others like it) perform an annual ritual in which they sacrifice one person. The intent and/or motive of the sacrifice is not articulated, but this is part of the point of the story. This ritual has gone on so long, it is done out of tradition and perhaps still adheres to an original intention (sacrifice to the gods, etc.), but it is also done out of mindless habit. The tradition continues for one or all of these reasons: symbolism, religion, tradition, and habit. But your question suggests that the ritual might have additional significance.
The notion of ritual is nothing new. And the sacrifice of a scapegoat goes back to Ancient Greece, Ancient Syria, and the Bible. A scapegoat is a person who is sacrificed, banished, and/or blamed for the sins of the community.
Mathematically speaking, in order for this ritual to continue, the community must produce enough children to continue the tradition. People will die of natural causes and so on, and there is the additional sacrifice each year. For every person that dies in the town, at least one more must be born in order for the town to survive.
So, the children are vitally important because without them, the lottery does not continue. This is an odd notion, but in a twisted way the lottery is an incentive to perpetuate the community itself. The lottery morbidly encourages people to marry and have children so that it and any other rituals continue. Since it encourages having children (to ensure the lottery continues), it is like a fertility ritual. In this strange and indirect way, the lottery is for the children because it provokes the community to have children. This may be why the author says, "The children assembled first, of course." The future of the town and the lottery depends upon children being born. They become the lottery. The lottery can not continue to happen unless children are born "first." Children are born into (and for) this tradition, brainwashed from birth.