Professional football players exist in a situation we can't really understand. It's transactional (draft, sign, re-sign, release, cut, trade, etc) but to be successful you have to work together well as a team over relatively short periods of time (a season) before disruptions occur.
Coaches have to get players on the same page and buy in. There's different ways to implement it. Carroll has a way of doing it and that's where the "culture" description comes in. He's clearly inspired a lot of players during his tenure over the years. The dude uses the words but also comes through. You see people try to mimic this in the non-sports world, but it often goes poorly because they don't know what they are doing are aren't acting in good faith.
When I hear people talk about how "culture" doesn't matter, or more broadly disparaging NFL players because of how much they get paid ("JUST DO YOUR JOB"), I wonder what life experiences they've had. With rare exceptions, NFL players still have to buy into something that is greater than themselves to be successful.
Whether you're a player for the Seattle Seahawks, an aircraft technician for Alaskan Airlines, or an infantry lance corporal in the Marines, leadership and teamwork matter; that's the "culture". When people just start acting completely transactionally and showing up for a paycheck things start breaking down.
