I believe it's a factor, but not the dominant one.

Me and my SO were considering a third for a brief moment, but it was the amount of living space and our age (35+ for the mother) that ultimately made us decide against it.

I was totally in for getting a new vehicle BTW.

Interestingly I have two siblings and we had a serious and expected downgrade as a family in living space just when I started attending grade school - a smaller apartment than mine currently.

As for the car seats the regulations came in when my younger sister was in 3rd grade or so, so he just decided to wing it without the seats.

Families of 5+ have very few options in terms of cars, among them the Peugeot 5008 - reviled both by reviewers and owners alike.

The loss of front bench seats has been a terrible blow - a six seat sedan has capabilities that can fit a large number of families (the oldest is often big enough for the front) but they’re rare as hens teeth. In fact, offhand I can’t name a current model with one even as an option.