I’m not sure how you arrive at the idea “it’s empirically false.” We know from multiple IPSOS/MORI polls that Brexit supporters and Trump supporters are less educated, more obese, more ignorant regarding current affairs, have lower incomes, etc. than the average citizen. Meanwhile if we look through the archives to see who supported Hitler, Stalin, et al most eagerly prior to their establishment of surveillance-based rules of terror we likewise see that on average those people were also less educated etc.
Today we like to pretend “everyone is equally intelligent” and invent all sorts of “intelligences” to whitewash the fact that in reality it is possible to arrive at meaningful definitions of intelligence and to measure that quality across sizeable populations. But pretending doesn’t make anything true. And when we analyze the appeal of Trump, Brexiteers, Hitler, Stalin, et al we find that they always (i) kept their messages very simple, and (ii) pitched them at a level suitable for less intelligent people to understand, and (iii) repeated the core themes over and over and over so as to permit the slower folk in the crowd eventually to latch on and cheer along with their peers.
These facts, which are a matter of public record, suggest to me that empirical evidence supports my position more than it supports an alternative interpretation.