But at least the way I read it, Bennet is saying that the NYT has a duty to help both sides understand each other, and the way to do that would be by giving a voice to the right and centrists without necessarily endorsing any faction

I think that this is a superficially pleasing argument but actually quite dangerous. It ignores that the NYT is itself quite powerful. Anything printed in the NYT is instantly given credibility, so it's actually impossible for them to stay objective and not take sides. Taking an army out to quash protestors gets normalized when it appears in the NYT, which is a point for that side of the argument, but the NYT can't publish every side of every issue. There's not enough space on the whole internet for that. This is why we have that saying that I mentioned in the other comment, that journalists should afflict the comfortable and comfort the afflicted, or that journalists ought to speak truth to power. Since it's simply impractical to be truly neural, in the sense of publishing every side of every issue, a responsible journalist considers the power dynamics to decide which sides need airing.

The author of the OP argues that, because Cotton is already a very influential person, he ought to be published in the NYT, but I think that the exact opposite is true. Because Cotton is already an influential person, he has plenty of places that he can speak, and when the NYT platforms his view that powerful people like him should oppress those beneath them, they do a disservice to their society by implicitly endorsing that as something more worthy of publishing than the infinite other things that they could publish. For literally all of history, it's been easy to hear the opinions of those who wield violence to suppress dissent. Journalism is special only when it goes against power.