The video that prompted this thread is the epitome of a true quick free kick in the attacking 3rd.
Attacker is fouled, second player gets the ball, ensures it's stopped and in the right place before quickly slipping it through to attacker 3 who scores.
For me a quick free kick is one where the referee is not having to get involved proactively, no intervention to get players away, maybe only a quick "yes" if the player says "can I play ref?" As a referee you need to ensure its in the right spot and also the ball is stationery when the kick is taken, but that's it.
The moment you've got to start managing the situation by shouting to get players away, or manage treatment for players then the opportunity to go quick is lost.