I think McCoy is essential, as the human-emotional foil to Spock; the classical dynamic is of the three of them interacting. The others are more like "people on the bridge (or intercom) all the time but who don't do as much." Scotty feels like he should be more important but I can't think of what he does.

Mind you, most of my Trek comes from novels, which can actually give 'minor' characters more time, but can also focus even more on the big three.

Of course, there was that Kirk/Uhura kiss (allegedly meant to be Spock originally but Shatner demanded the role).