I agree with the OP, and anyone going by reason instead of folk wisdom / traditionalism will agree as well. No player, none, ever, won the Stanley Cup by himself. The "great players are what make winners, if you're good enough you'll win" argument is wrong - no one is that good, the "winners" are the ones fortunate enough to be on a great team.
"So (amazing player who won a cup) was just lucky?" No, of course not. No one is saying that. That's a strawman. Their greatness contributes to the cup wins, but so does having 20+ good teammates. It's a combination of factors. The fact is that no matter how good you are it's possible to be on a team bad enough that you won't win the cup, and no matter how bad you are it's possible to win the cup if you're on a good enough team, so cup wins mean nothing. Anecodtes about great players who also won cups prove nothing, just like the fact that I can name Marcel Dionne and Kris Draper doesn't prove winning the cup makes you bad.
Your ability + the rest of your team's ability = team's success. X + Y = Z. The higher X is, the higher Z will be, but if we don't know what Y is, we can't say that high/low Z is proof of a high/low X.
Trottier is right in so far as watching a player's playoff performances and judging them is very different from counting cups. But a player who was great in the playoffs but couldn't win because of his team (Hasek on the Sabres) shouldn't have that held against him.