Mission Impossible and I Spy were mostly action shows and I'm sure MLK appreciated that African Americans were treated fairly on their shows, but I think he especially took note of Star Trek, because it took place in a future that reflected his own vision of the future where prejudice, at least amongst humans, was largely ignored. Also, it was African American woman and not a man being a recurring character on the show. Just ask Whoopi Goldberg what an influence she had on her.
I do get what your point that the story has changed details over the years and I do hope it did happened and wasn't bullshit. I remember telling Nichol's story when I was in 2nd grade when we were learning about MLK. I first heard it on the William Shatner's Star Trek Memories VHS tape. My teacher's jaw dropped and gave me an applause and the other students joined in. It reminded me of Star Trek VI.