Considering generations in the past, and still in a lot of poor countries have raised multiple children in the smallest of spaces, I just don't believe there is a causative relation.
Much more plausible to me is the association to women's education, causing women to delay their first pregnancy and opt out of further children. Also there seems to be an association between general misery or fatality and birth rates.
While it is of course possible to raise a lot of children in a tiny space, it is not something you necessarily want to do. Especially if no-one in your environment is doing it.
It is not really a happy life if you have to spend your evenings in utter silence because there is a baby or a toddler sleeping in the same room two meters away.
All these arguments assume rational arguments, especially economical, are the most important factors in deciding to make children.
I believe that irrational factors are easily overlooked but predominant. There is a certain biological drive to raise children, stronger in some people compared to others. And various factors modulate this drive, again unconsciously.
Most of the time, in free societies at least, it is the woman who makes most of the decision. If she really wants to have a child, nothing and nobody can stop her, short of a reproductive problem.
Much more plausible to me is the association to women's education, causing women to delay their first pregnancy and opt out of further children. Also there seems to be an association between general misery or fatality and birth rates.