But listen to any Herb Alpert recording, and then listen to any synthesized trumpet. The gulf is as wide as the Atlantic. (Part of the reason is that Alpert plays with the notes and the transitions, while a synthesizer just emits a note.)

Well, maybe a crap one, or one not designed for that. Trumpets have some convenient physics that are (comparatively) simple to model, the problem you're thinking of may have more to do with the control system (i.e. not based on actuation of valves, but skipping right to the oscillators) than some inherent weakness of synthesis.

(I played a trumpet as a kid, so I suppose my ear for it is more discerning than most. My talent proved elusive, but the lasting value was I can appreciate how incredibly good Herb Alpert was. He could play like Fred Astaire could dance and Ernst Udet could fly.)