“I’ve always been under the impression that it was a student grass roots movement against the establishment.”
I was in high school then and thought the same as you. My teachers in the bay area applauded the students for protesting.
The Free Speech Movement at Berkeley was led by a well-organized group of young socialists (SDS). They were trained to create “agit-prop”. There was zero spontaneity about the student protests, although the students themselves were tools of the Left.