O'Neill was on record as saying it was all about the quality of delivery. He was about 70% correct in my opinion. And in those days we had the likes of Young, Barry, Petrov, Milner, Downing who could all deliver the ball accurately and at pace. If the delivery is not good, then it really doesn't matter about runs, movement, positioning etc. Having said that, when the delivery is of sufficient quality, players better know where to be, where to go, timing of the movement etc. The addition of this set-piece coach has emphasized the quality of delivery, but also has added that extra 30% to ensure players know what to do once the first ball comes in.....
If you look at the Bailey corner from Saturday, Digne panics and feels like he needs to get his head onto the ball as there is a Villa player (Dougie I believe) steaming into the space just behind him. If that run wasn't happening he could very well have left it for the keeper.