While I agree that it seems like a cold way to look at it, you have to look at it objectively. If you're playing in a game where scores are being kept, and tournaments are being played, then the goal, is to win. If you're not as good as your mates, and they need the best players on the ice, then you need to recognize that it's not you, and take one for the team for the sake of potentially winning, even if it means you lose a bit of ice time. It's not about you, it's about the team, and being a dick at a crucial time in the game, is not something that a team player does.
You're not the only one paying fees, so why do you feel like your opinion is the only one that matters, when the rest of the team wants you off the ice for that particular play, and have paid equally as much money individually? If you don't care about winning or losing, then go play drop-in or pick-up, and stay out of league play, because you're ruining the fun for guys who like to compete, win, and push themselves to play their best.
This isn't about whether the ice time was equal or not. This is about the fact that it is a team sport, and not an individual sport. One guy's want, does not supersede the team's need. You're one voice on the team, not the only voice. If you're outnumbered, then democratic process dictates that your way, is not the way to go. If you refuse to go with what the majority wants, then you should go play golf or something, because you're selfish, and are completely missing the concept of teamwork.