You are taking issue with the first 2 minutes.

It's the friggin setup to the movie.

sounds like you have a problem with intelligence being inherited.

... which is a eugenisist idea.

The whole premise of the film in the first two minutes states "smart people aren't having kids while dumb people are" and the result of that in the film is a society of morons (who, also, somehow have automation and high technology, which requires, you know, some amount of intelligence). There's not really any other way one can take the film other than "we need to make smart people have lots of babies and prevent dumb people from not having kids lest we end up with a society full of morons" which is, by definition, "unnatural control and sterilization" (eugenics).

I mean saying "you take issue with the first two minutes" is a little like when a racist makes a racist claim and then you say "ok, what do you think should be done about it" they just say "I'm just sayin'" because they don't want to actually say the thing out loud. If you make a satire which uses as the premise that "smart people don't have kids, while dumb people have lots of kids" and then depicts a future that is degenerate, full of stupid people doing stupid things for stupid reasons, it's hard not to take that as advocating for eugenics on some level by saying "we need to make smart people have lots of kids and prevent dumb people from having too many kids."

Literally, from the wikipedia page, here's the ending: "Joe discovers the time machine was an amusement ride, a detail Frito was aware of. Joe becomes president and marries Rita, with whom he has three children. Frito becomes vice president and has 32 children, all stated to be "the dumbest kids ever to walk the earth" in contrast to Joe and Rita’s children, who are "the three smartest kids in the world."

The two people from "the before times when everyone was smarter" have three of the smartest kids around and the dumb guy has 32 of the dumbest kids around. It's not being subtle, this isn't even something you would find in high school English for close reading. It's literally saying "smart people have few kids and dumb people have lots of kids and this is bad and will result in a degenerate society that we must avoid. Hell, for a film making this claim, it's not even written that well. It's more "what if Cleetus from the Simpsons ruled the world" than "what if the global IQ dropped because smart people didn't have four kids a piece". Does make fun of Anti-intellectualism? Sure. Does it do it in a way that actually points to the problem? No! It blames breeding rather than the actual reason for all the anti intellectualism.

The reason for all the anti-intellectualism and stupid bullshit in the world right now isn't because smart people aren't shitting out kids at a rate of knots. It's because of right wing authoritarian people, who themselves have at least some brains, manipulating the public discourse for their own selfish aims. Cases in point:

• Climate change denial: Climate change deniers are literally claiming Scientists don't understand what weather is and it's being parroted and pushed by right wing publications who, I might add, aren't being run by morons, but people who will benefit monetarily from us fucking the planet.
• Campaigns to ban abortion and contraception: We literally have several countries that are showing why banning contraception and abortion are bad ideas (Ireland and Romania and the people who are campaigning to ban them are making claims like "Condoms don't stop AIDS" and "the moment the sperm hits the egg, the zygote is fully aware". All of this is being promoted by people who aren't morons, but by people who will benefit monetarily from banning contraception and abortion from all the cheap labour.
• Campaigns against transgender rights: TERFs are literally developing a fake history around transgender people, saying that it was both due to the Nazis and only invented in the 1980s, which their own propaganda (The Transsexual Empire, a book that advocated for the complete elimination of transgender people from the population, was published in 1979) disputes. The people who promote this aren't morons, they are just extremely hateful and benefit from using violence to enforce strict gender norms and the culture war that distracts people from the real problems of this age.
• Campaigns for child marriage: Yes. In the USA, the country that is the sole focus of Idiocracy, people are advocating for CHILD MARRIAGE. And they're not merely doing this because their fucking pedophiles, they're doing this because they benefit from girls not getting a proper education, and people having kids when they can't afford so they become beholden to low paid, long hour jobs to pay for their kids, which you know, benefits organisations like Dollar Tree and WalMart and Amazon who rely on cheap labour who can't afford to say no. All of this is not even bringing up that having a child at 12-14 is often times deadly.
• Campaigns to abolish the Department of Education: This is a part of Project 2025, the reason is very simple, and it's not because people are stupid: They want quality education to be only available to the rich while out of reach of the poor because a quality education allows people to make more money and be politically active and be harder to manipulate. The people advocating for this aren't dumb, quite the opposite, they're smart and are going to benefit financially from this.

If I were to summarize the it all, it's not that stupid people are in charge, it's smart evil people using anti-intelectualism to get their way. To drive it home, during the Brexit campaign, the former Education secretary, Michael Gove, when someone someone was interviewing him and challenging him on brexit, stating that every single economist was saying Brexit was a bad idea, Michael Gove said "people are sick and tired of experts". That's the issue here: we're being told not to actually listen to people who know what they are saying and listen to right wing ideologues who are either benefiting from this or being paid by people who are. And again, Michael Gove isn't a dumb person, he's an Oxford graduate, he's just a really fucking horrible person who, amongst other things, despite being Scottish himself, made a televised diatribe where he claimed Scottish people were poor morons who didn't deserve self determination.

And I know that's a lot of text, but sometimes you have to go into detail. So here's the tl;dr:

1. Idiocracy's whole premise, start to finish is "smart people don't have as many kids and dumb people have too many, and that will result in a degenerate society of morons."
2. The real cause of anti-intelectualism are terrible people promoting bullshit, fighting against scientists and even just generally moral and ethical people, for their own self interest.