Anti-intellectualism is a byproduct of meritocracy
Trying to combat anti-intellectualism with education will make meritocracy worse and cause even more anti-intellectualism.
American anti-intellectualism is caused by our economic system. If you have a high paying job you have to resort to anti-intellectualism to justify your pay. If you have a low paying job you have to resort to anti-intellectualism to “own the libs”.
But it’s worse than that. As Quinn Slobodian details in Hayek’s Bastards: Race, Gold, IQ, and the Capitalism of the Far Right large segments of Americans were done with intellectualism as soon as climate change was realized. If I’m paid a lot to destroy the global environment then the nature of merit must change to justify my pay — even logic itself must change.
The financialization of America also greatly contributed to anti-intellectualism. In explaining the 2008 global financial crisis Alan Greenspan said
I made a mistake in presuming that the self-interests of organisations, specifically banks and others, were such that they were best capable of protecting their own shareholders and their equity in the firms
In a meritocracy self-interest must be worshiped above all else. There is no room for intellectualism, because questioning the value of individual effort must be forbidden. Anything requiring collective action, climate change, regulating banks, research, etc., means individual organizations and leaders within those organizations are actually part of a society and not deserving of tremendous individual merit and pay.

Ascribing merit to an activity immediately starts a slippery slope. Raising children has merit but then the world population exceeds 8.2 billion. Science has merit but then it invents nuclear weapons. Education has merit but then it kicks off an arms race causing higher and higher degrees to be eligible for the same job.
Then anti-intellectualism follows along that same slippery slope as people decide to believe their activity still deserves merit long after it stopped being useful. The natural enemy of anti-intellectualism is honesty not education.
Honesty is a skill that can be trained but not one that is taught by our educational system. In fact the intersection with honesty our schools offer is constantly under attack.

Nor does an education focussed on STEM imply training in honesty anymore than any other kind of education. Scientific integrity is its own course that you don’t get just by learning science.
Republicans are not trying to defund education because they fear it will make Americans intellectual. That’s just a smokescreen. The real reason for Republican attacks on education is just the usual racist desire to keep marginalized people poor.
More important than education for promoting honesty is creating a more healthy environment for US citizens.

Dollar for dollar the best way to promote better mental health, as explained in the classic book The Spirit Level, is to reduce wealth inequality IE mitigate the meritocracy.
Then there are those that would blame anti-intellectualism on rural America:
rural identity in particular views experts and intellectuals as an out-group
Unfortunately it’s a fact that experts can be just as anti-intellectual:
In their book Merchants of Doubt, science historians Naomi Oreskes and Erik M. Conway state that Seitz and a group of other scientists fought the scientific evidence and spread confusion on many of the most important issues of the 20th and 21st centuries like harmfulness of tobacco smoke, acid rain, CFCs, pesticides and global warming. Seitz said that American science had become “rigid”, and his colleagues had become closed-minded and dogmatic. According to Oreskes and Conway, Seitz used normal uncertainties of scientific evidence to spread doubt about the harmfulness of tobacco smoke.
The Seitz they refer to is Frederick Seitz, “4th president of Rockefeller University from 1968 to 1978, and the 17th president of the National Academy of Sciences from 1962 to 1969. Seitz was the recipient of the National Medal of Science, NASA’s Distinguished Public Service Award, and other honors.”
That’s the problem with confusing educated and honest — the two have no correlation.