It’s not farfetched.
Many, or most cars (not suv) does not not have a place for three carseat in the back.
It’s the case for us. So unless we buy another bigger car, which we cannot afford, our family will stay with 2 children.
Another solution would be to wait until the oldest is more than 10 years olds, so no car seat will need, but we will be old at that time.
It’s not uncommom. Many families, when they have 3 children have to buy another car. The problematic is even much more critical for families who have 4 children or more. The car become very difficult to find, very expensive, very costy for inassurance and gazoil and overtaxed by the governement.
We also have to note that in France the bigger the car is (or heavier) the better the governement give us monetary mallus. And “familial” vehicule that allowed more than 2 carseat are in the category that are overtaxed.
I agree that it often is an excuse for someone that do not want more than 2 children. It is for my husband who wish a bigger car to put more objects, but not for having another child. And in our case, the two bedroom house, the difficulty to moove (as we are owner) and my husband unwillingness to moove is another big parameter. The children financial "burden, even if justified is another excuse. There are many others excuses such as an “unable” mother.