The intro to Idiocracy makes the point well: https://www.youtube.com/watch?v=hL1-340ODCM
It's kind of a joke, but also really not. It's frightening.
It is a joke.
Idiocracy relies on the assumption that the "more intelligent" people are snobs and too busy and too perfectionist with their life to have children, while the "less intelligent" people are mostly having sex and are too lazy to apply contraception properly.
Essentially, the first half of the movie is a justification for elitism.
I'm less afraid of Idiocracy to become true, but more afraid of politicians acting as if they believed in this movie.
The second half of this movie is more interesting, though. It may be interpreted as an analogy of how the most gifted people (in the sense of "intellectual giftedness") may feel living in today's world. (by putting today's "normal" people into a dumb world) But even that is, of course, a funny hyperbole, and nothing to take too seriously.
The joke is in the exaggerations and charicatures. The truth is what is real that would cause this joke to be made in the first place.
Obviously smart does not equate to "snobbery" or "perfectionism". And less smart does not equate to "more horny". It is a comedic layer added by the directors to make you laugh, because it is a comedy.
But the truth beneath the funny is that less smart people generally make poorer life decisions. For many people in many contexts, less children for a longer time is a smart life choice, and more children more quickly is not. Because more children more quickly is, in most contexts, in the category of poor decision, it tends to happen more with less smart or educated people.
I don't see how it is an argument for elitism at all. It's an argument for promoting education.
I see it the same way, as an argument for promoting education, throughout the whole population, not just some "elite".
However, I've also seen people arguing for the opposite direction: That the "elite" must be protected from the "mob", must be socially more separated than it already is, or it will die out. And that it must become more attractive especially for the "elite" to become children, while the "mob" will take care of themselves. (The argument wasn't phrased with exactly those words, but the sentiment was essentially that.)
The culprit, however, doesn't seem to be raw intelligence as such, but modern western education, which seems to have a dose dependent sterilizing effect on females. In that light, high IQ isn't fertility-reducing in itself, rather low IQ is protective against modern western education.
Also generally, more clever people focus more on quality rather than quantity, and babies are no exception. Norm are 1-3 kids, not because women are not fertile anymore (unless climacterium is reached), rather time & resources are considered.
Even this is very simplistic & dumbed down view on these matters.
I admire the way you phrased this.
The difference in size between educated and uneducated families might be very large in societies that are in transition on the development scale, but I wonder if the same is true in a society that is stable on that scale. Do uneducated families in the West have more kids than educated families, or do they just have their 2.1 children earlier in life?
First, you have to avoid conflating culture with genetics. No one 'evolved' to be a good engineer: there has been an insufficient amount of time and probably selective pressure for an engineering gene to arise, spread, and flourish. Instead, we've developed an engineering culture, which can readily spread independent of the genes of the individuals currently carrying the engineering culture. Indeed, even today after a century or two since the dawn of modern engineering there are many fantastic first-generation engineers.
Second, I don't really believe that modern civilization is deficient as an environment for breeding intelligence. Fundamentally, intelligence isn't excellence at a particular culture or being supremely well suited to a particular niche, intelligence is adaptability. Intelligence lets us survive in a great variety of niches, that our ancestors never had the chance to evolve for, to learn new skills and develop new cultures.
Unlike the static worlds of yesteryear where generations could go by without a change and you could expect to live a life identical to your parents with (maybe) different names for the principals involved, everything changes now constantly. Culture, technology, jobs, food, language, social mediums. Everyone has to deal with it. People may not need to demonstrate their intelligence in a way that pleases a certain subset of the intellectual elite, but they're being required to adapt -- by their peers, their mates, their potential mates -- on a scale that has rarely if ever existed in human history.
Besides which, it's fictional. While there is some truth to the fact that "less educated" people tend to reproduce at a comparatively higher rate than the more educated, it's a fallacy that somehow this is resulting in a "dumbing down" of the human race. For example, raw IQ test scores have been increasing over time quite dramatically (IQ is always normalized to the mean so 100 IQ is by definition the average).
Idiocracy is an entertaining story, and it holds some lessons worth taking to heart, but it is not the truth.
So, I'm agreeing with you in a strict selection sense.