... If we're seeing the story of King Arthur, we expect the sword in the stone and Merlin and Morgan le Fey and Mortdred and for the Grail Quest to come up at some point, and so on.
So I guess the question is, does Trek as a concept have any organically-grown narrative points of reference like that, things that people have come to expect alongside the bare-bones concept, costumes, technologies and general somewhat-more-ambitious-than-pulp sensibility (assuming people even agree on all those things as necessary to the concept)? I think the answer to that is probably "no"
Right, if the next Star Trek movie has Sargeant Crunk and his crew explore the ocean in their submarine, the USS Interpreter, you can sit through it. Star Trek is humans, warp vessels, teleporters, and exploring new worlds, seeking out new life and civilizations, boldly going where no man has gone before.