I don't think it's the _only_ important thing, but I think I do agree with the sentiment that it may be the most important.
Personally I'm all about diversity of perspectives when it comes to building a team. I would be thrilled to have both a principal who loves jira, and one who doesn't. Someone who tries to straddle both just strikes me as neutered.
When it comes to roles, I always end up back to the Ron Swanson quote "never half ass two things; whole ass one thing."
I do agree with you though, one-size fits all definitions are almost all laughably based on "vibes."
Then again, isn't management pretty much just vibe cultivation?