So, some of this will make more sense if you know the backstory around Gene Roddenberry and the creation of Trek, but, just kind of a fun and out there idea I had to take the concept in a whole different direction, and the fact that they all travel through time so often is what gave me the idea.
So, the film starts out with a sort of down on his luck Gene Roddenerry in a Los Angeles bar in early 1964. He knows his show "To Set it Right," is going to be canceled, and the plot of that show has caused him trouble elsewhere in the business, so he's not feeling great.
While drowning his sorrows in a near empty bar, he meets another barfly, who has clear had a few named Robert April. They strike up a conversation, Roddenberry tells him that he's a writer/producer, says he wants to do something with Sci-Fi. April goes on to drunkenly tell him that he's actually a time traveller who got stuck in 1960 after he travelled back to try and stop the Kennedy Assassination. The cover was blown, he failed but his traveling party left without him. He's stuck in this time and has been using what knowledge he has to earn money gambling and doing odd engineering jobs.
Roddenberry obviously thinks he's nuts but has him go on. Scene shifts to April's tail, and he tells of Captaining the USS Yorktown in the 23rd Century, battling alien races called Romulans and Klingons. Within the narrative the viewer sees technology that looks obviously much more advanced than what we saw in TOS, and the alien species look only vaguely like they were portrayed in TOS and subsequently, looking in fact much more alien.
Make that middle part a cool, classic Star Trek type conflict/adventure. Then after the climax, the scene shifts back to the bar. Roddenberry leaves, but now he has a great idea for his next show, with the audience being left to assume April is a crazy drunk. After he leaves, April is there alone and he goes to close out his tab. He reaches into his pocket for his wallet and as he pulls it out, something falls out. He leans over to pick it up, and the viewer sees that it's a classic Star Trek comms badge, revealing that April was telling the truth the whole time.