As cut in both "The Cage" and the "inset" footage for "The Menagerie", it does indeed appear Spock is employing some sort of "proximity" controls the change the screen displays. A wider shot left on the cutting room floor reveals Spock is actually gesturing to an "assistant" sitting at the science board to press some buttons or flip some switches. (This is the same crewperson, a brunette wearing a ponytail, we see when Pike extracts his printout report.)
I know; I know; if it's not in the final print, it's debatable. Just sharing a tidbit some members may not know.