Bernstein's vernacular has little precedent. Here's a movie composer writing in essentially American idioms, shunning styles well established in Hollywood. Jazz wasn't really foreground movie music yet. What's more, Bernstein paved the way for subtle orchestral scoring, using players only as dramatically needed. No gushing violins when a solitary flute could say the same thing better.
Drums start the score, trombones and trumpets follow.
It's this big band sound that stays with you. Little footnote here. During a break in recording STAR TREK V for Jerry Goldsmith, I heard a percussionist play a familiar riff on his snare. Right on cue, a trumpet player came in. Everything was spontaneous.