Doesn’t the constitution also say something about things like attempting to overthrow the government, assisting foreign military forces in attacking domestic targets and taking bribes from foreign governments?

A more equal society than the most equal to have ever existed (in modern times at least - some hunter-gatherer tribes are highly egalitarian)? No, I don’t think so, but the kind of issues Americans popularly hypothesize as reasons for low birth rates are just not relevant in these societies, and I don’t see lower inequality having much effect on the real reasons people have no or few children there.

The least unequal societies all have low birth rates though. While inequality is still a serious issue in these societies, I’m not aware of any evidence or mechanism suggesting addressing this would increase birth rates.

And how do you renewably generate the power for electrolysis without solar or wind farms?