This is an outlandish and ridiculous hypothesis with zero substance to it. All research points to it being the other way around. They higher the economic pressures and the less freedom people have in reproductive healthcare, the higher the birth rates. The moment Germany introduced the birth control drugs, the birth rate dropped.
If a country develops from the level of Somali to something like Germany and the birth rate tanks to somewhere slightly above 1, increasing the birth rate by maybe 0.1 by enabling more personal decisions has literally zero impact.
Of course I'm still a proponent of decreasing economic pressure on parents and enabling reproductive freedoms like pre-implementation diagnostics. The consequences on birth rate is just something we have to deal with one way or another.