Seeing the Amish in the Techno-Primitivism Episode reminded me of something I used to wonder years ago: Will the super-religious dominate the world in a few hundred years?
To show the power of compound growth, the Amish population in the United States in 1920 was 5,000 and today it is about 350,000. So they've grown 70-fold in one century. And that actually understates their birth rates because various estimates have 10-20% of their children leaving the community to join "our" modern society. So they have higher birth rates than indicated above. There's apparently no stigma to that either; children are given the free choice to remain in the community or move out, with the vast majority electing to remain. There are similar Anabaptist groups such as the Mennonites and Hutterites with slightly different, but mostly similar beliefs. They live all throughout the Americas including Argentina, Bolivia, Mexico and Canada and their birth rates are high everywhere. If their combined world-wide population is approximately 1 million today and they double their population every 20 years then the Anabaptists will surpass 4 billion people in the mid-23rd century. I always thought that was a ridiculous & unrealistic number until I saw the Ecumenopolis episode!
Normally when people make long-term demographic predictions they turn out to be very wrong because the world is always changing. For example, people used to think Catholics would dominate the United States because of their no-birth-control rules but the very low birth rates today in USA, Spain, Italy, Poland, says that people aren't really following that anymore. And a lot of those countries have aging populations.
So how does this play out in the future? Do the super-religious "techno-primitive" societies dominate the Earth in the future? Possible arguments against this:
(1) Future technology makes life so easy that most Anabaptists decide to join the mainstream society. This seems reasonable except that they currently reject modern society because they believe hard work is essential to a strong character. I'm not sure if even more advanced technology will make them change their minds.
(2) They decide to embrace birth control to keep their numbers down. Again, this seems unlikely because it violates their core beliefs. And if one sect embraces birth-control they'll simply be outnumbered by those that continue to reject it.
(3) They're more vulnerable to catastrophe. Maybe some event in the future will greatly reduce the human population, but disproportionately towards primitive societies. It couldn't be a famine though as we would simply help them out. And this one doesn't seem likely either because most of the population-collapse scenarios I can think of would disproportionately affect a high-density, high-tech urban population and, thus, wipe us out rather than them. I think a low-tech rural population is quite resilient against catastrophe.
(4) Life Extension technologies will disproportionately benefit the mainstream societies and offset some of their exponential growth. Interestingly, though, they can still theoretically out-populate us by a wide margin even if we become immortal and they do not.
(5) Expensive farmland. This is what I think will ultimately hold down their populations. As the rest of the world grows exponentially wealthy many people will want to buy large rural estates. Low-tech religious communities will find it difficult to compete for these lands and will eventually run out of suitable land for their lifestyle.
TL;DR: I do think that super-religious "techno-primitive" societies will become a large proportion of the Earth's future population but they cannot completely dominate because they will run out of farmland and will not be able to out-compete wealthy urbanites that wish to buy expensive land in the countryside. However, my guess is that this won't happen until they achieve populations in the hundreds of millions, or possibly billions.