Intellectualism is public by definition. There are no "private intellectuals". There are private scholars, but intellectualism denotes not just scholarship but engagement with the world. The public intellectual is a redundancy invented in America and the American university, denoting a Moses coming down from the mountain, either the mountain of Academia or the mountain of the isolato.

Farrell

It’s an argument that King himself would probably wince at - he seems too much of a steak and potatoes guy to want to describe himself in such grandiose terms. Even so, I think the description fits. There’s a strong case to be made that his books and stories, taken as a whole, tell you more about the Matter of America than the work of any other living novelist. And they are not only deeply intelligent but politically intelligent. If you want to know what the US was really like under George W. Bush, you’ll probably find out more from reading Under the Dome (which is not even one of King’s best novels) than Ill Fares the Land. The ease with which a slick rightwing populism can slide into something approaching fascism. The ways in which community loyalties can sour politics or redeem them. The intertwining of politics and petty personal jealousies. King gets it all. He has both an understanding of American life that Judt (for his many intellectual gifts) lacked, and the ability to express that understanding in clear, unornamented prose that can speak to millions of people.

"The Matter of America" links to a Shalizi reading and viewing list. It's culture for quants.
If King is a better writer than some give him credit for it's only because he's transcended the culture that Farrell and Shalizi are a part of.

The 4th comment links to Dylan Riley, which is something at least. I get a lot of hits for that link, which doesn't say much for its popularity.  Judt was a moral conservative, a liberal only in context.

Farrell defending democracy, above

[Judt's] disdain for popular communication goes together with a version of social democracy that emphasizes the ‘social’ at the expense of ‘democracy.’

And in the past

I’ve suggested that academic freedom is a good thing on pragmatic grounds, but also made clear that it fundamentally depends on public willingness to delegate some degree of self-governance to the academy. If the public decides that academic freedom isn’t working out in terms of the goods it provides, then too bad for academic freedom.

Farrell doesn't understand democracy; he doesn't understand culture, let alone intellectual culture.

Individuals, society, the state: they are three different things.
Plus ça change,
Shalizi