re: Why didn't Karate Kid ever carry key characters over?

It was how movies and sequels were done in the 80’s.

If they could get actors back, great. If not. Who cares? Write a new story without them.

Unlike today, back then, sequels aimed more often to do something new and creative with each movie and then maybe sorta tried to wrap up the larger arc (if there was one) in the last film of their contract. No biggie. People weren’t near as hard core on the serial nature of story telling back then.

See the high lander movies