I disagree with you on the dump and chase. It's actually easier. I play both D positions and have for a long time. When the puck is dumped on you and your on your offside, when you pivot 180 to chase, at that point your on your strong side. I'm a right D, on the right side of the ice (when skating towards my goal) going to fetch a puck in the right corner. When I get there, I'm on my forehand and can put my ass between the puck and the chaser. When I'm playing RD and pivot 180, I'm on the left side of the ice. It's harder to protect the puck. It's harder to rim it on your backend.
What's a lot harder playing off side and requires you to be quicker is guys trying to blow by you. Playing the right, when you pivot, your stick is between your body and the goalie. When you play the left and pivot, your on your back hand to try to prevent the drive to the net. That is often not possible. You're footwork is different. You can't gap as well. And you're more vulnerable to the winger cutting to the middle.
Brannstrom was getting hit a lot because he hadn't adapted to the speed of the game imo. I don't think D men get hit harder or more frequently because they're playing off side D.
Disagree and I've read about this, played and have observed at the NHL level.
When you are on your off side after your pivot on a properly timed dump in at point of retrieval between the corner and behind the net you do not have the luxury of going high and off the glass (or low) to the strong side with a strong side turn post pivot. On your strong side your pivot can take you into a turn that puts you in a position to immediately advance the puck up that side, or seek the centreman who if back quick enough can provide an outlet.
A left shot on RD is forced to either A/ turn to the weak side and play it out on a backhand. This relies on a good turn otherwise the forechecker will telegraph it and be in a good position to lay a potentially damaging hit. Someone like Chychrun is able to handle this to a better extent because of that quick turn. Not unlike Sanderson he is a good skater that can elude that pressure. That said, it still doesn't allow you to get to the forehand and make a proper exit pass, your still left distributing or clearing from your backhand which is not ideal.
Option B is to swing it around to your D partner and all you're doing there is putting him in an almost equally vulnerable position where the only option is to pretty much keep swinging the puck around and quickly force it out of the zone because the opposing winger is anticipating the pass and closing the gap on the LD just as fast or faster. You're essentially telling your partner to deal with it and that's how weak pairings get hemmed in.
This is all magnified by being small and not a great skater. On the off side you are at a severely disadvantaged position and I'm talking well after the pivot and with a forecheck that has timed it and placed the puck in a way where you're not able to squeeze them out with your backside the way you are saying.
Added to that, you would have coaches that would exploit the matchup and be sending their physical guys in on that play, the Maroons, Wilson's and Reaves of the league. Early in Brannstrom's career I think he would have gotten punished if he was forced to deal with that weak side pressure over and over. These aren't weak hits that are easy to avoid either, these are take a hit to make a play situations, hard squeeze hits or full on plastering.
And yeah, I agree with what you're saying about defending the 1 on 1, entirely, I get that. From a physical standpoint though, it's the dump and chase and the elusiveness it requires to avoid it which Brannstrom didnt have early in his career, or perhaps still doesn't.