The problem is that Star Trek isn't a very good science fiction premise.
Really good science fiction isn't really about the space exploration or the robots, or the time travel, or what have you. Really good science fiction uses those features to make you think about things that you might not otherwise have context to think about.
When you watch Star Trek, the original series especially, you really need to take it in with the culture of the time. It was a time when racism was normal, and women were treated as second-class citizens. Star Trek presented a scenario where men and women, whether they be white, black or not even human, were all treated by each other as equals. What made it especially classy, is that this was done without any characters getting preachy. They simply went on about their business, as well they should. To characters on Star Trek, working together in harmony was a given, and that a man should think women lesser than he, or that a white man should be somehow superior to a black man was more alien than the strange creatures they met. It was so many generations lost to their culture, that it was no longer even a thing. This is what Gene Roddenberry presented to us with Star Trek. In creating this series, he meant to show us that this is what we can become, if we abandon these notions of hate and inherent superiority, and just work together. Infinite diversity, in infinite combinations.
Roddenberry couldn't have expressed that in a modern scenario, nor would it have worked so well in an historical scenario. A military starship three hundred years in the future, on the other hand? It was a pretty damned elegant fit, and frankly, I think that makes it a pretty worthy work of science fiction.