Game Time Start After a Made Free Throw

Q: After a successful last free throw, the game clock should start after the ball is touched on the playing court following the throw in, and not after the free throw is successful - correct?

A: You are correct. Here is the article from the rulebook that supports your statement:

50.1 Started or restarted when: 

On a throw-in, the ball touches or is legally touched by any player on the playing court.