Historically, what you speak of is an idealized and generalized image. What village are you talking about? Where? When? What was the socioeconomic status of the family? Etc.

In reality it would vary whole lot, not just in terms of time and place in a general sense, but also for individual families. If you had many relatives nearby, perhaps, but in some cases you might not, or you might actually have to be taking care of not just your children but also your parents-in-law who are disabled and your aunt who is mentally unstable partially due to her own husband and children dying in the famine a couple years back.

And maybe you are also poor so you need to work land that isn't even your own, in addition to your own (maybe rented) plot, and you are socially shunned on top of that and your neighbors sure as hell aren't going to help out with your own children. But at least you only have two kids now since two died and you managed to give another away to live his whole life in a monastery.

I think kids and their free labor were the biggest wealth generating asset for the poor and as such wouldn't be given away except in the most extreme circumstances.