They could and it could be tournament (some world champ at hockey were like that, some soccer euro cup are like that), bigger sample size and so on.
But the strategy of making the Stanley Cup bigger than the nhl itself, is a good one and a nice distinction.
A big issue with the current nhl to make us care a lot about the president trophy, parity and the home ice advantage being virtually nill.... who care ?
Would it make you save a playoff round, assure you the home ice all alone in a context for which it matter, give you an bad first round opponent... ok