"Wouldn't the real cause of the depressed birthrates be the requirement to own a car in order to have children?"

Yes. The one-time setup costs for "properly" raising kids are probably around $30k. All the kids stuff is extra expensive (in the west) and for the kids seats you need a large car (in the west) and there's social stigma against kids sharing a room (in the west), so you also need a larger apartment.