This is a really neat concept! I have a few thoughts, but I'm going to really focus on how this would look in a D&D setting, and specifically how players and characters would interact with it.
I think one neat source of inspiration could be bees. Only one bee in a hive is giving birth, but all the bees work together to keep the hive functioning for future generations. Though 99.9% of the bees are not passing on their genetic material, if their traits and behaviors fit the environment, then it's more likely that more bees like them will be born. It's kind of like evolution through altruism.
I feel like a progeny-worshipping culture would work similarly. Their belief is that any child could be a good or messiah figure reborn. But those qualities will only be recognizable at a certain age (maybe 10 years old, just for a nice round number). To be safe, until the age of 10, every child is treated as if they could be a god.
In this society, those making important decisions are not the elders, but the children. As soon as they develop the ability to make decisions, children are asked to give advice on what crops to grow, what evils to prepare for, how to avoid plagues and other maladies. For pre-verbal children, their babbling and movements would be analyzed like portents and omens. For older children, there would of course be a host of adult advisers whispering in their ear.
Children who do develop magical abilities, but do not meet the other criteria of godhood, are still worshipped as saints or divine beings. So you could have sorcerers or planetouched characters who are treated as holy beings when they enter these communities.
Now this is D&D, so belief has consequences. In these societies, the children themselves can act as the material components for ritual spells. The children aren't sacrificed (in most communities), but rituals are enacted to cast Augury, Divination, Control Weather, and other spells. Of course, at 10 years of age, children can no longer be used for these rituals.
When characters enter a community, they will literally see children raised on a dais. Nurseries and school would be exalted places, carved with the images of the future gods. Children would walk on raised walkways above the mud and soil. They would be dressed in fantastic outfits, fed delicious treats, and adults would fall to their knees when a procession of children strolled by. Meanwhile, adults and the elderly would be toiling away, farming, building, and standing guard for the children.
When characters entered a palace, they would be dealing with a child ruler. At shops, shopkeepers would consult their children when characters attempt to haggle. Halfling and gnome characters may be mistaken for children themselves! Meanwhile, older characters, or characters from races that are traditionally long-lived (such as elves), would be treated like servants.
One more fun idea: these communities could have a practice in which nearly all births are scheduled for the same month. The whole month is filled with festivals calling to and worshipping these potential gods. It could be a festival of fortune-telling, luck, and rebirth. Fortunetellers and diviners from all over the realm gather at the center of this culture, and this of course draws adventurers from all over too! It could be a fun way to launch an adventure or campaign!
