I think it is a fair assumption Mitch gets down on himself in those times. He often says shit like "we need to stop feeling sorry for ourselves" and "I need to be better". Pretty confident he gets in his head in those times. Which is absolutely not a knock on him. It is a ton of pressure for a 24 year old.
As for him needing to be more physical. I don't know how you could say Mitch is any less physical than Kane and say Brayden Point. His timidness in the crucial moments has nothing to do with physicality IMO. It has everything to do with the time and space being that much less. He has less time to make those highlight real plays, and his line mates are experiencing the same degree of limited space. He tries to do too much when there isn't the space to do anything. He doesn't need to be any more physical, he just needs to simplify his game. The passing lanes are eliminated in playoffs. When you have the opportunity to put the puck on the net, you have to do so. Far too often in tight checking games Mitch tries threading a pass through a clogged lane and effectively neuters any scoring chance. Or he will try to make a play in neutral ice, that ends up going the other way off of a turnover.
Get the puck deep, and put the puck on net. No trying to make plays at their blue line, and no extra passes. Simple hockey. To say Marner is not good in and around the boards is false. He may never overpower you physically, but he protects the puck very well, and is really effective at causing turnovers. He is more than capable of playing this way.
Playing that way will inherently incorporate more physicality to his game because he will be forced to play in harder areas of the ice.