Interesting ideas, but also lots of assumptions and some of the anecdotal evidence seems to go against what you're saying.
E.g.: childbearing seems to be the lowest per person in areas with a well developed market system, like European countries or the English-speaking West. Why? More anecdotal evidence suggests fear: people are simply afraid that they can't afford a family, and the solo child is usually an accident.
Free market dogma also promotes costly private education that can only afforded by a minority of successful business savvy couples. And the same applies to private health-care. Meanwhile, the system is gamed by bankers, brokers, and other profit-seeking middlemen who profit by putting themselves in between the resources and the consumers of those resources, and artificially restricting access. If the ongoing feedback from the population isn't strong enough, such a system can trend towards Fascism, as seems to be happening in the US. How is that not "tyranny of the free market"?
I don't agree with any of this, particularly the underlying assumption that the US is a free market, but I won't even go into it because it's besides the point. None of what you're saying contradicts what I said. The Malthusian principle is not an "idea" or an "assumption" but a recognised and widely applied scientific principle. Populations always grow right up to the point when they can't grow anymore. Therefore, resources can never be over-abundant unless the population size is artificially reduced.
I'll reiterate:
-The wealthiest, "least authoritarian" economies somehow have the least ability to grow their populations.
-Developing countries, i.e.: the ones with less resources and more authoritarian governments, end up making more babies.
How is that possible unless there was either something you missed or something wrong with the theory? It seems Earth's population has been running up against hard limits ever since Capitalists figured out ways to profit from war.
Besides, why isn't Norway's oil wealth resulting in a population boom? Don't tell me it's because of their oppressive socialist policies that discourage families. That's why I'm suggesting that maybe exposing critical services like education and health to the amoral ruthlessness of "market forces" is actually more tyrannical than a forthright government that steps in and provides those things as public services.
There is no doubt that richer and more educated people have less children. That's a well-known sociological fact, and so it is not surprising that richer countries have lower population growth rates.
But global population growth isn't stopping. People figured out ways to profit from war thousands of years ago, and Earth's population isn't running up against any hard limits quite yet. A certain percentage of the world's population is always near starvation, then agricultural technology advances, allowing more people to be fed, and then the population grows accordingly so that roughly the same percentage is near starvation.
Are you saying that if resources were over-abundant (essentially, if everyone was arbitrarily rich), people would have less children? Perhaps, but I doubt it. A considerable percentage of the world's population would have even more children. You yourself said that the main reason people have less children in developed market economies is fear that they will not be able to afford a family. If resources were abundant, there would be no such fears and anyone could have an arbitrarily large family.
I would add another piece of anecdotal evidence: highly educated people in rich countries tend to be more focused on career rather than family. They delay having children to an older age, so as not to interfere with career development, and then end up with few or no children. This will also disappear in a utopian technological over-abundant society - people would essentially have nothing better to do than start a family.
Either way, in the long run, there are only two possibilities: either people choose not to have children and the human race will eventually go extinct, or (much more plausibly, as I said above) population will grow to the point where resources are no longer over-abundant.
The third possibility, that population size will remain constant, seems impossible in the long run. It requires that the global average number of children per couple will remain exactly 2 forever, and there is absolutely no reason for this to be exactly so (unless central planning is involved).
As for exposing critical services to market forces, that's a completely different debate...