A discussion about winning games is long overdue.
There is no math here, no numbers, and barely a cursory mention of statistics. The fights between "stats guys" and "other fans" are illusory. The real question is whether one is willing to take a measured approach to reality, i.e. figure out what's actually happening on the ice.
A hockey team wins the Stanley Cup by winning more games than it loses. Sixteen in the playoffs, and usually more than 41 games in the regular season. The road to the Cup is paved with wins.
A hockey team wins games by scoring more goals than the opposition. You definitely want to score more goals than you allow in any one game. Generally speaking you would also like to score more goals than you allow over the course of a season. Maybe there is something to be said about timely goals, or scoring enough goals to win in close games while simultaneously getting blown out in games that you were going to lose anyway. On the other hand, it probably doesn't matter. The road to wins is paved with goals.
How does a hockey team score goals?
The coach's perspective
Scoring more goals than you give up is a hard thing to do in the NHL. Some players are obviously better at it than others. Naturally, as a coach, you want to limit the ability of your opponents to outscore you. Generally speaking the players you are most concerned about are your opponent's best. Generally speaking the tools you want to deploy the most are your best.
Conversely, a coach is always concerned about his weakest players, the ones at the bottom of his depth charts. These players are the most terrible at outscoring, and he must hide them from quality. The blade cuts both ways however; a savvy coach salivates at the thought of dominating the dregs of his opposition.
Most bench management tactics arise from these simple notions. Power-vs.-power, building the gameplan around your gifted offensive forward, shift after PP, icing management, etc. all stem from coaches trying to gain a material advantage through matches or mismatches. Usually the coaches are very concerned about who is on the ice for the other team, but considerations about how far along the opposition is in their shift (the notion of tired legs) as well as which zone the puck is in (the notion of territory) are also important.
That's the bulk of the coach's contribution to outscoring*. Managing his players, not putting them in positions to succeed individually but in positions for the team to succeed as a whole. And it matters, the distribution of ice time is not at all even. Some players will, on the whole, get ice time that is harder to outscore in. Other players will, on the whole, get ice time that is easier to outscore in. Obviously their results vis-à-vis outscoring will be different, how could it not?
* The notion of "systems" gets a lot of press but there is not much innovation between coaches to be honest. On the other hand, the chasm between the best and worst hockey players is wide enough that a coach that doesn't work his matchups properly is actively hurting his team.
The player's perspective
The player is the coach's greatest and only tool for outscoring. Thus, the player must outscore. However, scoring goals does not just... happen. There is a process.
Players have skills. We can name several for skaters - strength, speed, puck-handling ability, vision, anticipation, shooting. For goalies, similarly, except add twitch reflexes and memory.
Ultimately as a hockey player you want these skills to translate into goal scoring so you want to be in a position such that you can shoot the puck into the net without the goalie being able to make the save. So we speak in the language of scoring chances. Gaining access, with the puck, to the "scoring area" as it is often called - the area between the faceoff dots and the posts up to the hash marks or maybe a bit farther out. In this area you can shoot the puck and if you make a good shot the goalie physically doesn't have time to react. Guess right and you look like a hero. Guess wrong and you look like a tit.
Everything that hockey players do is related to scoring chances. Face-offs, puck battles, take away passing and shooting lanes, create passing and shooting lanes, reverses, breakout passes, gaining the line, holding the line, cycles, etc. etc. ad nauseam. These are all in the name of gaining access and limiting your opponent's access to the scoring area. The notion of territorial advantage comes up a lot and it's easy to see why - out-accessing your opponent is easier to do if you have control of the puck in the offensive zone more often.
It's a pretty obvious truth that some skaters are more capable than others of finishing their chances. As well it's an obvious truth that some goalies are more capable than others of stopping opposing chances. But volume of chances is a huge driver of success, perhaps the principle driver over long time frames given the narrower spread of finishing/saving talent that exists now.
Role matters too. Coaches give out ice-time to youth like candy, it's always sweet. And the exact reverse for veterans. Outchancing is harder to do when you are facing good players that are better at denying you access to the scoring area. Similarly if you're always counted on to retrieve the puck from your zone, well chances are you'll spend more time watching it go to the scoring area (on-a-stick-off-a-stick-in-your-net) than you'd like. And you're not going to have as easy a time going to the scoring area yourself, what with it being 175 feet down the ice with 2-5 opposition players in the way. And plus you'll be tired after spending 25 to 30 seconds regaining possession of the puck, you're only human after all.
If your role dictates that you're going to have a harder time outchancing, then you'll surely have a harder time outscoring. You might have a harder time scoring, period. And on top of that, luck has the final say (see below). Doesn't mean you're not helping your team win, after all someone has to play the tough minutes and nobody has the goods on Lady Luck. This is the nightmare scenario for hockey players on a Canadian team. This is how local sports personalities start suggesting that you get demoted to a lesser role (who will play the tough minutes then?) or traded for magic beans.
A measured approach to reality
There are two keywords here: "measured" and "reality". Too often the focus is on "measured". The label of "stats guys" is a bit unfair given that the casual fan can quote reams of stats for his or her favourite players and teams. The term "advanced stats" is a misnomer; there's nothing advanced about shots, scoring chances, face-offs and time on ice. These are basic and essential elements of the game.
Statistics aren't the end-all or be-all, but reality is. Reality dictates that the game is played a certain way on the ice. We develop some numerical measures of how the game is played on the ice. We call these "statistics" but they are really just rink-side mirrors, reflections of how individual skills are translated into team wins. Most "numbers guys" are actually "reality guys".
A note on Luck
It's a bitch.
When we talk about winning games, we talk about a series of processes here, translating individual skills into wins and ultimately Stanley Cups. And at each process luck has a say.
Skills -> Scoring Chances -> Goals -> Wins -> Stanleys
At every step luck takes away sustainability and predictive value. The higher you get the more you lose to luck. The casual fan's focus is on Goals, Wins, and Stanleys (and sadly, not in that order). Which is fine. But when Stanleys are used over Scoring Chances as a reflection of Skills, that's when you get into trouble.