The real questions are therefore: how can we educate those people or how can we remove them from the voting pool?

Ultimately democracy can't exist where there are "those people" versus "us people". Lincoln's famous speech spoke of government "of the people, by the people and for the people", which only means something if "people" is a unified whole. Not "for us people, but not for those people". After all, the American civil war was about the inclusion of a different kind of "those people" in the voting pool and the abolition of the 3/5 compromise.

National unification is incredibly hard work, but worth trying.

"We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills" -- Kennedy

(this applies very much to the UK as well, but I'm targeting a primarily American audience here)

But the question is if "pure" democracy is what you should be aiming for. It might be a flawed concept, just like a pure free market is considered flawed.

I didn't say anything about purity, which is always an impractical concept. I did mention unity, which is probably just as impractical. "Inclusivity" maybe.

The important things about both democracy and the rule of law is that people are willing to acquiesce to decisions not in their favour because they recognise the fairness of the process and have a voice in it.

It's a hard problem.

I mean, there were once poll taxes and literacy tests, which were discriminatory. But how would one test for skeptical rationality? And how would that ever get implemented in a populist democracy?

Edit: typo

An indirect approach could be to figure out if sensitivity to fake-news/populism/echo-chambers correlates with other behavior, allowing you to select.

For example, suppose that a correlation exists with short-term greediness. In that case, you could alter the voting process such that the voter can choose between a small reward or voting.

This could also be justified by saying that politics should not lean towards short-term policies, and hence people who are prone to such thinking should not participate in the voting process.

Doing that for the politicians might be a better idea there. Too many MPs and Congress members only think about the short term effects of their policies as is, and it's certainly causing issues in the political world at the moment.

If you consider yourself not to be one of "those people", you're being very optimistic.

Well, self-awareness is the first step ;)