After reading your post, if I were you, I'd settle for What Are Little Girls Made Of?, Catspaw or That Which Survives.
What Are Little Girls Made Of? is IMO the only great episode of these but the other two has some elements that might appeal to your dude.
Catspaw? As an introductory episode? I would strongly advise against that, it's not at all typical of Star Trek and also, typically ranked near the bottom of ranking lists, I believe.
I don't like it. But I think that it could be perceived as more fast-paced and adventurish, which she says that he likes, and might be one of those episodes who most likely will be able to lure him into the world of Trek.
I can tell you that the first episodes of Trek I saw were those on The Best of TOS DVD (as a kid I only saw TNG and VOY (which I never learned to like) on TV):
1. The City on the Edge of Forever
2. The Trouble with Tribbles
3. Balance of Terror
4. Amok Time
Trouble With Tribbles, I've never liked nearly as much as most do, but, I do agree with the others in that list as great episodes.
Another really popular episode I've never cared a whole lot for is "The Giant Flying Terd" one, The Doomsday Machine
. However, I do like "The Space Hippie one" The Way to Eden
a great deal, which I think is fairly unpopular. So, yea, I see your point, but, I really can't see Catspaw
as being a good introduction epsiode to get someone interested in the Series, and if it does get them interested, I'm not sure it's typical enough of Trek to allow other episodes to be satisfactory, if more of that's what they're looking for