Does your church have a congregation-participation babtism where everyone stands up and promises to raise the child to love God?

When you joined the church, did you make a promise to support the church, which of course includes the children?

If so, then you have an obligation to interact with children to the extent that such interaction brings them closer to God.

Games like the ones you describe do this by making those children feel more comfortable in church. They assure them that they are loved by not just their parents but by their church family.