The vast majority of euros do grow up playing the other side, and it is an adjustment when they come to the NHL.
The reason they commonly play the "off side" is the difference in the ice surface and how it dictates the game is played. Watch international hockey and you will see it almost resembles a basketball or soccer game where the idea is to keep it away from the center. Because of the width (and depth of where the blue line is), in international hockey there is far less chasing players and board play, its a more zone style. Thus the advantage is to have your shot in towards the center of the ice. And in fact you can see this in north american hockey during a PP, where the setup is much closer to the international game (less board play and chasing more zone style D), and, guess what, the D pretty much always switch for a PP in the NHL.
The issue during regular play is think about how your body is positioned going into the boards if you are chased/hounded. If you are in the d zone, it means your head is facing back towards your own end (not up the ice) and if you want to chip it out, its a near blind backhand attempt. Be on the correct side, its a forehand where you have fully visibility. Same in the offensive zone, you go to the boards, you lose visibility and the forehand is actually facing your own net.
Yes some players do play their off side. Many play significant time on the PP, or are lefties who are just stuck playing the right because of the lack of RHD in the league (or just care about scoring). But it lowers the odds of making good plays in board battle situations.
That all said i am a bit torn on this issue. I do not believe Branstrom can reach his full potential playing behind Chabot, he is not that defensive stalwart, he needs PP time to be effective. Despite the issues, he may still be better off on the right hand side.