See, so this is the case both between different nations, and within the same nation. More spending power = fewer kids.

That correlation turns around at the lowest income levels; you get paid per kid. Yes it's not really enough, generally, but if you feed them nothing but gruel and get public benefits for everything then it's actually more profitable to pop out more, especially if you can get 20% from each father if they're plugged in enough to the system that most of the fathers are paying most of the time.

It doesn't turn around, it stays the same (except for different reasons on different levels of income ladder). Poor people have the most kids, worker class, fewer, middle class, even fewer, and high middle class, fewer still. It might not be the case for the very rich (top 0.01%), because they literally don't care about expenses, and can afford trophy wives 30 years them younger so fertility isn't an issue either.

How could it possibly stay the same when those at the lowest rung are getting additional benefits for having kids? Starting at income '0' they can only go up from there.

I readily admit the benefits are shit and only profitable if you're basically feeding them gruel and giving them little else, although sadly there are people who do this. If those at '$0' are the most fertile then they're (what initially could be described as paradoxically) going to have more income on average at 1+ kids due to increase in benefits, which necessarily turns things around at the lower end.

That's a misconception. This correlation exists even in places that don't have this kind of social security. It even happens between countries, or even historically.