Cenk Uygur's argument he's eligible for the Presidency

[I spent the winter writing songs about getting better]

I'm kind of interested in what some of the more legally-minded people here seem to think. Most people seem to be rejecting it and frankly if it was valid someone would've probably brought it up by now. But Cenk is arguing that despite being a naturalized citizen originally from Turkey that he is eligible and the natural born citizen clause of eligibility for the Presidency was overturned by the Fourteenth Amendment, specifically this part:

All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

The issue I see is that it states that a STATE may not discriminate against naturalized citizens, but this clause is in the federal Constitution. So based on these grounds it would be unconstitutional for any state to require that its Governor be a natural-born citizen and disallow naturalized citizens. But this clause in the federal Constitution is not covered by that.

So how right am I? Any other issues involved?

[I spent the winter writing songs about getting better]

Am I correct though in that this means that no state may institute a natural-born citizen requirement for any office? I'm not sure if any did even prior to the 14th Amendment, but this seems to be the one implication from Cenk's argument that holds any water.

[I spent the winter writing songs about getting better]

It is a clever argument and it is one that an unscrupulous Court which cares about an evolving law might have adopted once upon a time (it feels like the sort of thing that Breyer might've liked, actually), but especially given that the Congress which passed the Fourteenth Amendment considered and decided not to revise the natural-born citizen requirement, I can't imagine the current Supreme Court adopting Cenk's interpretation.

(It would also be a weird and retrospectively unfair decision, since many individuals who might've otherwise run for the Presidency and perhaps been elected chose not to run because of this particular clause. If it is repealed, it should be through an explicit amendment, not through the end-run of SCOTUS reinterpreting what everyone spent 200 years reading in a particular way.)

I doubt it's even the conservatives on the court, I'd wager any decision on this would be 9-0. At most Sotomayor would write a very sympathetic concurrence in which she implies about how this requirement really really ought to be repealed for most of it but then concluding at the end that it hasn't been.