Faceoffs are important, but for sure if there is a pecking order it is towards the bottom. It's true value is probably unknown though, since it's not a variable that's been isolated. A winning team could be bad at faceoffs, but we don't know how much better they could have been if they were good at faceoffs. Same goes the other way with bad teams who are good at faceoffs.
However, it's not about compromise. Contenders are just great in all areas. Being bad in faceoffs is a deficit, and a team with faceoff superiority can leverage that into wins. It can often be hidden in the regular season, but in the small sample size that is the SCF, you want to have good faceoff ability.
To me, what you need to have to be a true contender is all-in-one players. When Chicago won cups with Toews, he was their best center but also great at faceoffs. For us, if our coach sometimes puts out Letestu instead of McDavid only because of the faceoff, then we've got a problem. It shows that until McDavid becomes great at faceoffs, that's a deficit we will have.
In fact, faceoffs can be isolated a lot easier than many other aspects of the game. People have looked at thousands of events and have come up with compelling information on the impact of winning vs losing a faceoff on both possession and in terms of goals for/against.
You are correct though that what a player can do after the puck drops is far more impactful than winning or losing a draw. Now if you are one up or one down with 7 seconds left, winning a draw may have greater importance. But those situations are rare, and still the event itself is like a modestly weighted coin.
Going back to 2010-2011 here are the playoff face off numbers for the final four: *=Champ **=Runner up
2010-2011:
Tampa 49.5% (8) Boston 52.0% (3) *
Vancouver 49.9% (6)** San Jose 53.1% (1)
2011-2012
NYR 48.4% (15) NJD 47.2% (16)**
LA 49.7% (10)* Phoenix 51.4% (6)
2012-2013
Pittsburgh 49.7% (8) Boston 55.3% (2) **
Chicago 46.8% (14)* LA 48.2 (11)
2013-2014
Montreal 48.3% (11) NYR 47.3% (12)**
Chicago 49.6% (10) LA 52.9% (2)*
2014-2015
Tampa 48.0% (10)** NYR 46.5% (12)
Chicago 53.1% (3)* Anaheim 52.7% (5)
2015-2016
Tampa 47.9% (15) Pittsburgh 50.5% (7)*
St. Louis 51.1% (6) San Jose 47.1% (16)