is there ANYTHING we can do to keep those ill-informed believers from taking over the country?

Yes, read Blowing Smoke.  I wrote a lot about these effects--confirmation bias, cognitive dissonance, etc.

First of all, this article is overly simplistic. It's not the case that all convictions are like mutant supervillains that absorb contradiction and become more powerful than ever.

People do tend to resist challenges to their deeply held convictions, and some people go to great lengths to establish elaboration rationalizations in the face of contradictory evidence, but it all depends on the individuals and the context.

One of the keystones of the rationalization mechanism is that it often relies on affirmation from other people. On some level, we know when we're rationalizing, and one reason we try to convince other people of our irrational beliefs is to get them to affirm those beliefs. That's why you sometimes get hysterical explosions of irrationality that spread like a religious cult.

But that means that challenging people's beliefs can also disaffirm the rationalizations. When Joseph Welch and Edward R. Murrow told people what a schmuck McCarthy was, they helped to reversed the hysteria, and McCarthy fell from grace even faster than he had risen to prominence.

So if you tell any particularly deluded members your family that their beliefs are wrong, it actually has an effect. You may not see it because they will try to resist the challenge by rationalizing that you're a crazy liberal, and if you remain the lone dissident, you won't get very far. But if siblings and cousins were to join in on your side--particularly those whom they cannot easily dismiss--it would wear away the resistance of the hold-outs.

So what we need to do is keep speaking the truth and aim to turn the tide.

I've adjusted my diet ...