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.
I've got some bad news for you about, well, pretty much all of human history...
For example, if miscarriages are criminalized, and access to birth control is restricted - both real things that have been attempted or have actually happened in the US as a part of anti-choice policies - the only safe choice is to not have sex, ever. Which means you're probably never going to have kids, instead of before where there was a chance you'd get pregnant and then decide whether to have the child or not. Now it's too risky to even have a chance of getting pregnant if you have no autonomy. I certainly would never risk it in a state with anti-choice policies.
The intent of these policies might be to raise the birth rate, but I'm not sure they're going to do that. We'll see, I guess.
IMO the demographic crisis is more likely to be influenced by other factors, like the rising costs of raising children, the increasing constraints and pressures on parents, etc. But those policies don't help.
The decision to have kids should be a deliberate commitment between the parents, not some kind of lottery where one falls pregnant then decides what to do next.
It's better not to fall pregnant at all otherwise