I watched the show as a kid during its original run on NBC and I never remember being confused by Spock or his presence on the show. He was the cool, super-smart alien guy with pointed ears and green blood, who was all about logic. All of that was explained on the show, and by my dad, so I just took it at face value.
It was a sci-fi show, set in the future, so, of course, there was an alien on the show. And it's not like good-guy aliens (or robots) were all that shocking a concept. Remember, most of us sixties kids were also watching reruns of The Twilight Zone, The Outer Limits, My Favorite Martian, and The Adventures of Superman ("strange visitor from another planet . . . "), not to mention movies like
The Day the Earth Stood Still, Forbidden Planet, It Came from Outer Space, and so on, all of which were airing regularly on our local TV stations around the same time STAR TREK debuted. And those shows and movies were not obscure cult items known only by hardcore sci-fi fans; they were popular entertainments enjoyed by general audiences.
Having an alien on the bridge of the
Enterprise was no more jarring than having Robby the Robot on the bridge of the starship in
Forbidden Planet, or Flash Gordon teaming up with Prince Barin on the planet Mongo.
And that's not even counting all the sf books and comics many of us were reading back then. Trust me, Mr. Spock was not more jarring than the Martian Manhunter or Aquaman or the Legion of Super-Heroes.
(And comic-books were not a niche thing back then. You didn't buy them at specialty comic shops, catering to hardcore fans. You picked them up at drug stores and news stands and flea markets.)