Learning/studying football is like learning/studying famous games of chess, or mathematics (at the fringes where new things are happening). It's loads of fun, but it takes years to be able to do on your own, and it's much more beneficial for fans like us to have an expert walk through plays/concepts/strategies (that's why JT O'Sullivan and Brett Kollmann have such popular youtube channels, and why Cover 1 is awesome)
When you watch NFL game film, it is almost always possible to break down what you see into at least variations of concepts you recognize - Beasley/Davis running split level route patterns, LBs playing deep in Tampa-2, offensive line shifts called out by the QB, whatever. NFL teams are constantly stacking changes onto these things which makes it hard sometimes, but I like to think that football games usually consist of largely-well-performed strategies that play out according to the skills of the most important/relevant players involved in the play, coaches having a strategic edge through creativity or game planning or in-game adjustments, and a dash of randomness. When things go wrong, you can usually see why, and imagine how proper execution in that role would make things work again.
Hockey is a lot different, you can't watch it this way. Studying hockey is more like judging different types of music or art than studying Bobby Fischer. You can know all of the different types of forechecking, PP/PK strategies, whatever, but 70% of what you see in the game can't really be cleanly classified by those things, and the network of what "systems" and "styles" work well or poorly against each other isn't nearly as well defined as, say, finding the hole shot against Cover-2, over routes against single high, screens against blitz etc. It's messy and beautiful. Coaches always talk about how they draw up plays during timeouts at the ends of games, and have basically never seen it happen as drawn up.
The best way to learn what you are seeing, learn what players are good and why, is to watch a lot of hockey. You will develop the ability to meaningfully distinguish between players and teams, and will be able to tease out interesting nuances over time. As ill-defined as hockey is, that makes the well of things to notice infinitely deep. There are posters here who are good at this sort of thing, I can tell you will be one of them if you stick around. But the difference between most teams on either side of the good/bad delineator is pretty small and takes dozens of games to flesh out, which is why they play so many games and why you always read that "we need to see 20 games before we know what this team is". Every "advanced stat" you mention, save for the very best and very very worst teams and players, ranges from about 47% to 53%. You want your team to be slightly better than their opponent at creating chances, shots, and limiting them for the other team, and stopping them in net, over long periods of time. This will get you to 50 wins and in instead of 40 wins and out, or 30 wins and drafting top 5.
The coach's job is not to develop advanced new plays that attack the weaknesses of a defense, to study another team's offense and find ways to stop it - well, that is part of his job, but he will never drive his team's success through this medium the way an NFL coach can. This sort of skill does increase in importance in a playoff series, when you see the same team over and over again, but to get to the playoffs it's more important that a coach can keep his team focused, together, playing cohesively, while balancing chemistry, rest, and preparation over a grueling 6 months. Part of this is "film study," but sticking Cozens against McDavid in 2021 to get a 3-2 win was not mathematician Granato, it was gut-feel, psychologist Granato. The same decision might not work nearly as well 3 days later. What buttons can you push at just the right time?
You are asking very good questions and your eagerness and enthusiasm are just what you'll need to be an elite hockey mind. No real need to dig into the calculus though, because there isn't too much there (at least at a level we fans have access to)
I have come to watch hockey two ways - focus on individual skaters that I like or want to learn more about (this usually happens when i get a chance to watch extended highlights, or rewatch games, where you can pause and rewind etc.) and then focusing on space, how teams make space, where they are capable of making space. Where are the best places, what are the best ways to open up space. And how can you close up space on other teams, make it hard to get to the important spaces in your zone. There is no right answer for any of that, but you can start to tell over time if your team is good at it, and which players do it in what ways. Take a snap shot of the screen any time there is a decision being made by a skater on your team or the opposition - does he have multiple options, or is everything closed off? A turnover is bad, but the player making it didn't necessarily make a bad play if there are acres of ice between him and a teammate. When you watch Carolina and do this, almost all of the time they are in mini-odd-man-advantage situations where the decisions are easy and obvious, whether they are in their own zone, on the boards in the NZ, or in the slot. For so long, the Sabres, whether good players or bad, through tactics or no chemistry or for a million other reasons, were constantly being suffocated, nothing was easy, options were an exception.
I could ramble forever, but yeah, try not to go turbo-geek with hockey, just watch and enjoy how awesome it is and whatever makes itself available to you will come eventually. The game can talk if you are willing to listen, and sometimes it will let you in on some pretty cool secrets
Oh, and advanced stats are basically an acknowledgement that it's hard to quantify hockey, but we are going to throw whatever techniques we have at it anyway and see what we get. They're cool and interesting, and provide a lot of info since it's impossible to watch every team 82 times, but when you can watch a team 82 times I would trust what you see over what they say
Matt Stafford is probably a better QB for my comparison than tannehill. It would have been tantalizing. Agonizing. Just an example that highlights one side of the extremes
