I think the easy answer here is coaching. At the end of the day, it's the coaching staff's job to ensure the structure and style of play are in place and consistent. We all see it - when they are dedicated to their structure, they make it very miserable for the other team. They protect the goalie and limit high-danger chances. It's not hard to see how we lead the league in SOs (I believe).
The more difficult answer may lie in the tendencies of the older core. It's not the 3rd or 4th line that starts running and gunning and free-wheeling. Typically, it's Sid or Geno for a shift or two (based on the ebbs and flows of the game), and then it compounds from there. So it might be that a guy like Sid or Geno feels the need to break from the structure at times. Whether or not that's a natural response to needing a boost in the game, that tying goal, etc, who knows but I would suggest that when we start to not look great, it does tend to start with them. Then it looks like that self-destructive "Oh, I made a mistake, I need to fix it, and then get back to the structured game".
And just how much control Sullivan and the coaches have over that...honestly, it's hard to say. We know the political, no-bias, non-reality-based answer is "well they are the coaches, they have 100% control," but I don't think it's that easy. This is where I tend to wonder about the relationship and true dynamics of the relationship between Sid/Geno/Letang and the coaching staff. I don't see Sullivan having (whether he wants to or not) a super tight leash on them.