I don't think that a diverse crew is a necessity, but some opposition rooted in friendship like with Spock and Bones worked well. I am sure you figure something out to "spice it up"
If I may ask, if you intend to do it only for fun, wouldn't something like short prose, a prolonged short story or shortened novel of about 100 pages not also work? I don't wanna interfere, just ask because for reading fun a script seems a little bit odd for me. On the other hand I guess one can more quickly establish a character via dialogue than via long descriptions in prose, the advantage of drama.