YES, THERE WAS AN INTELLECTUAL CONSERVATIVE MAGAZINE

Albert Jay Nock and Francis & Helen Swift Neilson started the Freeman in 1920 and although it ceased publication with the March 5, 1924 issue, it did have very impressive contributors, including Lewis Mumford, John Dos Passos, Thorstein Veblen, Bertrand Russell and Charles Beard.

UPDATE: Nock was no Fundie, as you will learn from this passage he wrote in 1916:

The ideals and instruments of Puritanism are simply unworthy of a free people, and, being unworthy, are soon found intolerable. Its hatreds, fanaticisms, inaccessibility to ideas; its inflamed and cancerous interest in the personal conduct of others; its hysterical disregard of personal rights; its pure faith in force, and above all, its tyrannical imposition of its own Kultur: these characterize and animate a civilization that the general experience of mankind at once condemns as impossible, and as hateful as it is impossible.