TL;DR - Not completely unimportant. The most arguments can probably all be summarized on how much importance and weight should be put on things like face offs, physicality and intangibles.
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The issue with certain old and new school "focuses" that some have hinted at is that there are direct and indirect correlations that are important, but difficult to quantify in new school analytics.
A win is a win. But face offs like hits/gamemanship intangibles are indirectly involved with certain expected outcomes.
ie:
What is a loss of a face off? An uphill battle to gain possession of the puck.
Why is possession important? It is extremely statistically unlikely to score a goal without possession.
Why is less hits than the other team bad? More likely to sustain injury which leads to man games lost.
Why is more man games lost bad? Because you're more likely to be in an uphill battle to gain possession of the puck.
Why is possession important? It is extremely statistically unlikely to score a goal without possession.
Why is less hits than the other team bad? More likely to mentally place the players in a situation to lose possession.
Why is possession important? It is extremely statistically unlikely to score a goal without possession.
Now, you can argue that this logic is faulty. Sure. But I believe there are some that ascribe to this. I think we can also agree that even the most vocal against the importance of these facets of the game aren't arguing that they are completely unimportant, they are arguing the level of importance and the total weight that should be put on these facets of the game, because you can still technically win with only 30% face off wins. You can still technically win if you have 30-50% less hits against the other team.
I believe analytics is most interested in possession and attempts to score a goal vs total goals scored. It wants more specifics as much as possible. It also tries to quantify the momentum that old school loves. But to do so, it does have to simplify the understanding of the game and thus some might consider it flawed.
Old school on occasion (or often) looks at the situation in a longer chain and consider it more likely to start that chain in an efficient/effective manner if you win face offs, if you use some energy and effort to tire and slow and scare the opposition a bit, and admit you can't know or quantify if this strategy will even work or even back fire. Old school is OK with less specifics, but that doesn't help creating a tool to evaluating thousands of iterations of events to a high level of agreed conclusion from one person to the other.
The truth is somewhere in the middle. One day someone might be able to figure it out with an advanced computer script (I hate the misuse of the term AI).