I don't think anyone believes "culture" is completely unimportant, but it is completely secondary to talent, roster construction, and organizational depth. Those three latter things lead to winning play, which in turn helps engender a "culture of winning" or however you want to describe it.
Tom Seaver used to tell the story of when Gil Hodges became manager of the Mets in the late 60s, how he helped a team of young kids mature and learn to not accept losing after the really rough beginnings of the Mets during their expansion years. I don't doubt Seaver's account for a minute, that Hodges had such an impact on him and other players and helped them believe in themselves and to expect better from themselves, but the fact is that the Mets went from a near-last place team to a World Series winner in 1969 because they had players like Tom Seaver on the roster to begin with, top flight talent that was good enough to contend for a pennant thanks to the good work done to bring talent into the organization after starting out with very little in 1962. That's not to say Hodges' influence didn't matter; he's a legend to the organization for a reason, and what he likely imparted on his players was the need to demand the best of themselves, to not feel comfortable with losing, and to work hard and give it their best effort no matter what and trust that their talent would carry them to better days. It did, but it took time; Hodges demanding better of them didn't translate into wins until that championship year, because even talented young teams have to take their lumps before their glory days come along.
Plus, it wasn't as if Seaver, Koosman, Jones, and other stars from the Miracle Mets team had been there since 1962 (well, besides Ed Kranepool, who was basically the Daneyko of those early Mets teams) and experienced all the losing themselves, they all got there later on and only experienced a few low-standings finishes. We shouldn't conflate our experiences as fans of feeling mostly frustrated with the Devils since 2012 with the experiences of guys like Hughes, who's only in year two, and as long as the organization is demanding hard work from the guys then I'm not really worried about a "losing culture" mentality seeping in. Just watching them, you can tell they're trying, but that not being good enough is the nature of the beast when you're the literal youngest team in the league, even younger than some college teams.