IMO, power and speed are more important than strength in most sports. This doesn't mean that strength isn't important or that an athlete should avoid deadlifts. You can improve power, speed, and strength all at once.
Or, another way to say it, strength is a means, perhaps the primary means, by which speed and power can be improved in trained athletes.
Obviously, John Broz is a great coach, so I doubt he could have meant what he said literally. In my sports (BJJ/MMA/grappling), slow movements, and even "no movements" (isometric contractions) are a
huge part of the game. Example of isometric? Think about maintaining posture against a tough opponent while you are in his guard. Hopefully, you'll get moving and won't be there forever, but the fact is, if you can't be terribly strong isometrically/with minimal movement, you're going to get broken down, wrapped up, submitted, etc.
It would be nice to do everything smoothly, quickly, and explosively, but the matter of
the opponent makes that impossible. In grappling, you are frequently grinding away with all your might, and it is a simple physical reality that the maximum loads you can move explosively are considerably less than the maximum loads you can move, period. So, yes, maximal strength, slow movements/isometric movements come into play all the time in my sports.