I think fans if they actually knew how what percentage of players actually don't care *that* much about winning a Cup or not would be shocked. Shocked.
Because fans have a romanticized view of the game that they've stuck to as children, but NHL players aren't children, they know what goes on behind the curtain and have an unromanticized view on it in many cases.
There's a million things going on with these guys off the ice too, investments, mistresses, parties, family and friends non-stop hitting them up for "favors" (which is basically "I need money, can you lend me some"), individual goals, getting the biggest contract, lifestyle, proximity to wider family, what club you're going to hit up after the game, getting coach off your ass, being on a team where you have buddies you can go hang out with after the game, etc. etc. etc.
I'd say honestly maybe 35% of NHL players really want a Cup *that* bad. A lot of them are more in the "I'd like to have one sure" camp, not in the "I have to have one or my life won't be complete" camp. And honestly that 35% number might be being generous.