Yes and no. Look at Tom Wilson with the Caps, he was an instrumental part of their playoff success last year. Playoff hockey is tighter checking and more aggressive than the regular season so teams that have no toughness is going to find it harder come playoff time. Add in the fact that ice conditions deteriorate drastically once May rolls around puts speed teams at a bit of a disadvantage, not to mention the puck bouncing all over the place which affects the top playmakers. You still need toughness to get through 4 rounds of playoffs and the Leafs don’t have that, neither does Tampa Bay.
I do not equate "toughness" with fighting. There are big, tough, mean, strong players that do not have to resort to fighting to justify a seat on the bench. They play a solid game and that aggressive style of play and taking the body will win games. But if I am the owner and I am cutting their pay cheque? I ain't paying them to fight.
I also do not think ice conditions in May are so bad that it will eliminate the speed and finesse of the game.
When the NHL started to participate in the Olympics the fans got a taste of what true professional hockey was. Fast, smart, excellent play-makers, tough, aggressive, hard hitting...There were numerous articles and conversations about how amazing that type of hockey was. Eventually the pro leagues figured it out that that was where the future of the game was and it took 15 years or so to get there.