TOS didn't look futuristic even watching as a child a few years later in the 70s. It does have a very deliberate stylization that's great to watch, all the beehives, miniskirts, funky boots, sideburns and all that plus the minimal approach to the ships, tech, and controls. Trek always took you somewhere fantastic but a vision from its day and I don't think completely unintentionally (like Buck Rogers years later).
I think it came out just before that crazy leap into the 70's. Think how much different shows looked ten years later, it was a huge change. When the Hayes code broke in the late 60s movies and TV became very different. Movies from 1966 and 1970 seem 20 years apart not four. If it had come out post-Space Odyssey I think it would've been a different beast. Combine that with the novelty of color TV at the time and you get a very vivid crazy 60s future.
But I wouldn't have it any other way. It's part of what makes TOS great.