Doesn't the name say it all? This group was devoted to the life of the mind, and to complete collaboration and dialogue around all ideas pertaining to knowing what knowledge is.
So, yes, there was a lot of thinking about thinking here, which means this group never ran out of things to do. One major element that bonded all members together was a commitment to humanist thought. The Intellectualists believe that human beings have more agency than the institutions and social constructs that try to control them; they have faith that humans actually have some agency and self-determination; and they are okay with the idea that there is such a thing as "human nature."
The group was interdisciplinary, so why not have a brilliant Hungarian composer lead the charge? Bartók wasn't only a composer; he was also the founder of ethnomusicology (the study of folk music). He always had some little gem of wisdom, and he came up with an aphorism that—gosh darn it—Valéry himself wished he could claim as his own: "It is odd that the Bible says, 'God created man,' whereas it is the other way round: man has created God."
Čapek loved sci-fi before we had CGI, Philip K. Dick, and Ray Bradbury. Who else could claim to have written War with the Newts? By the way, did we mention that this is the guy who—are you ready for this?—coined the word robot?
Thomas was not just a novelist—he was an "intellectual novelist," and that made him a shoe-in for membership in the Intellectualists. He wasn't just about throwing together some great plot; he wrote books that reflected all of the crazy things happening in Germany—and in people's heads—in his era.
Mann actually came up with phrase "intellectual novel," which may not be as great as "robot," but it did qualify him for membership. If you want to see for yourself how smart this dude is, check out The Magic Mountain or Doctor Faustus. We're not worthy. (Okay, Valéry probably is.)