Bryan is not the first one to worry about schools. In 1962, John Holland Snow accused the educational establishment of subversion.
I believe that an educational movement or philosophy which minimizes or denies the possibility of our people and institutions for self-improvement -save through the expanding agency of government-is subversive.
It builds from there. Read the whole passage, which is found around page 7 (as numbered in the printed text), under the heading of "Opinion."
A contemporary libertarian might say that subversion is a good thing--we're supposed to be against government, right? But Snow was thinking of an America that embodies a libertarian ideal, and it is from that perspective he was against subversion.
Anyway, if you follow the link, you will find a book that takes a very conspiratorial view of how education evolved. I stumbled on the book during a Wikipedia surf while indulging my curiosity for how the Great Depression affected the intellectual climate. I recommend treating the book as one perspective, not as the way you should look at the world.
My view is that in the 1930's there certainly was a group of elite intellectuals who thought that people needed to be educated to the evils of capitalism and the virtues of collectivism. The question is what sort of influence that group had. I do not think they executed a complete takeover of the education system. I think that American education is an emergent phenomenon. It incorporates many strands. I don't think that the Marxist strand is everywhere dominant, but it is important in some instances, particularly in the way that the story of the Great Depression gets told.
I have a sense that the first draft of history of the Great Depression was heavily influenced by Marxism. It was a story of excess capitalist profit leading to collapse, and demonstrating the need for a collectivist system. I do not think this first draft ever lost its influence. In my view, this first draft has not been sufficiently examined or subjected to critical thinking by mainstream intellectuals. You have to go very far to the right on the political spectrum in order to find people who question that first draft.