How realistic was Karate Kid from a martial arts perspective in 1984?

It is not how martial arts were taught even back then, but it does reflect how martial arts was viewed by people at large. Eastern martial arts were considered noble and spiritual, and promoting peace. That's true enough, in a sense, but back then there was also very much the notion that with martial arts, the small and weak could dominate the big and strong. That is a very unrealistic expectation.

Moreover, the way Daniel is trained by Mr. Miyagi, is of course complete nonsense. But it very much falls in line with the romantic notions of martial arts that flourished back when this movie was made. There are plenty of martial arts legends wherein a teacher imparts martial arts knowledge by having an impatient and frustrated student do menial tasks. The student is unaware of the martial arts skills that he is honing, until one day he has a revelation. And so the student learns the martial skills, and he learns patience and respect. This has, of course, never happened in real life, but such legends have existed for a very long time - no doubt told by lazy teachers who were trying to get those nagging students to just shut up and do what they're told. In reality, there is no way to learn Karate from sanding the floor or painting the fence.