Well, at the time "Where No Man" was made, the idea was taken more seriously. In the '60s and '70s, some experiments seemed to show there was some kind of legitimate psychic phenomenon going on. It was later that we figured out that a lot of the experimental subjects were frauds; scientists merely report the evidence of their observations, so if you can fool their senses, i.e. give them bad data, they'll draw bad conclusions. Which is why it took magicians like James Randi and Johnny Carson to expose the frauds. (Carson and Randi totally demolished Uri Geller on
The Tonight Show by setting up their tests in ways that didn't let Geller use the cheats he normally used.) Also, a lot of it was bad experimental design by researchers who were maybe a bit too eager to believe. (Like Zener cards, those cards from
Ghostbusters with the shapes on them. Since there are only five of them, the odds of making a string of lucky guesses are much higher than they'd be for regular cards; also, some experiments failed to avoid reflective surfaces in which the testees could see the cards' reflections.)
So there was a lot of SF in the era that took psi powers seriously --
The Demolished Man, for instance, or a lot of Larry Niven's and Anne McCaffrey's work. (She was on record as believing that psi powers were real and scientifically valid; otherwise she wouldn't have used them in her work, since she always considered her work science fiction rather than fantasy.) WNM's idea that future science would have learned to codify and test for psi abilities in humans was pretty much in keeping with the thought of the era.
According to my DVD it was the one with Gary Mitchell that never aired.
Hmmmm....
Of course I could have misheard it and got things criss cross, so i will check it again to be certain.
Nope, it aired. It was the third episode broadcast. And I saw it constantly in the TV syndication package when I was growing up, so yes, it aired as often as any other episode (except "The Cage").
There is, however, a slightly different version of the second pilot that didn't air. It had a different title sequence with a different logo and a different theme built around Alexander Courage's Kirk motif, and it was edited slightly differently with a scene or two that weren't in the final version. Copies of it have been shown to the public at various times (the Smithsonian had a copy, as I recall, and it was probably shown at various conventions), but it was never shown on broadcast TV. That unaired version is available as a bonus feature on the 2009 TOS box set, so maybe that's what you're thinking of.