Genuinely asking this and not trying to be argumentative but people having multiple children is exactly what pushes the average amount of children up... so while in principle I get that the distribution doesn't matter, that is irrelevant surely as we aren't dealing with a finite number of children and then distributing them, we are dealing with the reality that if everyone who can have more children does so, overpopulation occurs?
We are not dealing with a finite amount of children, no, but anyone having children who wants them also doesn't mean everybody having children indefinitely, because most people don't want to do that. Most people only want a small amount of children, 1-3, many people don't want children at all. It is a rare thing to have a large family and so large families act as a statistical anomaly, being cancelled out by smaller families. Ultimately the decision of an individual to have one or two more or less children is neither here nor there. The average is what matters, and actually what tends to affect this most strongly are three things (all of which, thankfully, we have in the UK)
Low infant/child mortality
Access to contraception
Education for women
If you are worried about overpopulation (which arguably you shouldn't be, because there isn't a great deal of influencing to do, the population growth has already happened) you should look towards initiatives which improve these three factors in countries where the birth rate is still high.