It’s not even a problem of running into similar stories. Even the most similar of stories shouldn’t tread on toes as long as the characters are strong enough. You can redo City on the Edge of forever with Janeway if you like...she’s not Kirk, her love interest won’t be Edith, and you don’t have to use the thirties.
True, but the point is, you don't need to. I have seen what Gamers have done with the Trek setting over the years and look at the setting the Authors of Pocket Books have done with the Novels, or how STO's Trek setting has its own stories and style. Its a large setting and you can do so very much while "keeping in the lines" of the setting.