I dont even think that's necessarily true.
Colorado had the #1PP in the league earlier this year while having the 32nd ranked Faceoff% on the PP. Because the Avs are such a good zone entry team on the PP, losing that draw really isn't a big deal.
I don't think the faceoff itself matters one bit outside of very late game situations where you're up by a goal in 6 on 5, and then OT faceoff situations.
Those are the only two scenarios in a game where getting that puck possession especially matters. In 6 on 5 situations, it is significantly harder to get clean zone entries with the 5th guy on the ice for the team defending. And, losing that draw means the defending team could have a free shot on the empty net which could ice the game entirely. In OT, we see teams now recognize the value of puck control. You lose a faceoff in OT you might not touch the puck again for 2+ minutes.