We can see that now, because we have big-screen TVs and HD-quality home video. Audiences in the 1960s were watching on smaller sets, often in black and white, often with much less image clarity, often through fuzz and distortion due to poor signal reception or mis-adjusted knobs. (There were so many things that had to be synchronized or tuned just right to get a good picture, and they all depended on manual adjustment. You often had to work to get a decent picture, and sometimes "decent" was the best you could hope for.) So the substitution of the stuntman for the actor wouldn't have been nearly as obvious at the time -- which was why they were okay with doing it that way.
Trust me; this is exactly correct. As one who has watched TOS from the beginning. I never noticed the difference simply because you couldn't tell back then with the primitive equipment.