Besides Star Trek, I have played in and GMed other “ship show” RPGs that show the influence of Star Trek and react to it in different ways.
I was the captain in Distant Stars, a not-Trek exploration RPG that I hope to see published by its creator. I used some of my ideas for a no-budget-limits Trek setting as suggestions for taking some of the usual alien types further, so the wise psychic elders are insubstantial and the proud warrior race are seven feet tall with four arms.
In 2009-10 I ran The Stars On Fire, post Battlestar Galactica Military SF with a loose confederation of human colonies, some still in a cold war, united by first contact with an alien intelligence. The diplomacy and culture clashes were the clearest Trek-ish influence. (The creator of Distant Stars played in it.)
In both cases, we kept some features and dropped others. Both dispensed with transporters and universal translators, for example, and ran with different codes of conduct.
They both worked for players who weren’t fans of the series too.