There is lots of evidence that people in power are the most rational, but there is a huger prior to overcome.

Among people for whom power has an unsatiated major instrumental or intrinsic value, the most rational tend to have more power- but I don't think that very rational people are common and I think that they are less likely to want more power than they have.

Particularly since the previous generation of power-holders used different factors when they selected their successors.

Actually that's a badly titled article. At best "Rationality is systematized winning" applies to instrumental, not epistemic, rationality. And even for that you can't make rationality into systematized winning by defining it so. Either that's a tautology (whatever systematized winning is, we define that as "rationality") or it's an empirical question. I.e. does rationality lead to winning? Looking around the world at "winners", that seems like a very open question.

And now that I think about it, it's also an empirical question whether there even is a system for winning. I suspect there is--that is, I suspect that there are certain instrumental practices one can adopt that are generically useful for achieving a broad variety of life goals--but this too is an empirical question we should not simply assume the answer to.

The problem is that politicians have a lot to gain from really believing the stupid things they have to say to gain and hold power.

To quote an old thread:

Every politician I've ever met has in fact been a completely sincere person who considers themselves to do what they do with the aim of good in the world. Even the ones that any outsider would say "haha, leave it out" to the notion. Every politician is completely sincere. I posit that this is a much more frightening notion than the comfort of a conspiracy theory.

Cf. Stephen Pinker historians who've studied Hitler tend to come away convinced he really believed he was a good guy.

To get the fancy explanation of why this is the case, see "Trivers' Theory of Self-Deception."