I assume you mean what does that tradition speak of? Pretty much what Eliot says in the quote.
There's a positive and a negative aspect to the critique, in the sense of what it posits vs. what it negates. The positive aspect argues that the Good is accessible through something other than Reason. They're not against Reason, but see it as only part of man's heritage and not the most important part at that. This leads to a critique of modernity because, they argue, modernity doesn't acknowledge or allow for such knowing.
The negative aspect argues that by regarding everything as a technical problem, man becomes swept away by relativism, sees himself as the source of all values, and ultimately lacks any basis for knowing what is good. In this view, rational humanism is shallow and self-deceptive and Nietzsche, the prophet of man choosing his own values, is the great distiller of modernity, the one who took it to its logical conclusion. But where Nietzsche professes to celebrate that conclusion, people like Eliot and Weil and George Grant see it as a reductio.
An interesting thing about the thinkers in this tradition is that for a counterpoint to the Enlightenment view of man (which they reject for its triumphalist emphasis on Reason) they turn not to the anti-Enlightenment Romantics but rather to antiquity, before the split between Reason and Spirit (to use old-fashioned terms) occurred. You can hear this in the Eliot quote. Weil's lodestones were the Iliad and Plato. She, by the way, was the younger sister of André Weil. That was one amazing family.
