I mean it's always easier to keep the puck in on your dominant side, but you shoot worse and pass worse unless you have a strong corner player, which the Rangers don't.
It affects you in many parts of the game. You have all board play. You need to turn the back towards the ice and work the puck with your backhand in every battle along he board from the attacking blue line to your end wall.
But the big problem is defensively. In your own end. Imagine that you are Smith and play a shooter like AO. You are cutting off that shooting lane with one hand on the stick and your backhand. The NHL game is so fast and the marginals are so small. You will just bat at a much lower percentage in all those very important plays.
Then you have he whole situation where you fetch pucks in your own end. For many coaches it’s a big item to point out which Ds if any on the other team that plays the wrong side. Ship it in on them. They turn the back towards the ice, and play the puck with their back hand.
31 teams, 186 Ds in action at any given time. ONE, I repeat ONE of them thrives on the wrong side if you look at it with any kind of longer time frame. Hjalmarsson. It’s Chicago that plays a certain why and was very very synched with the rest of the team and did a great job. In PHX, he has had the same issues.
Go back 15 years and you could talk about pros and cons because the game was just slower. But it had still started. Have to go back 25 years almost before it was really up in the air...