I did the same thing moving from an IC to a management role and wholeheartedly agree that now I have the ability to influence the culture within my team of developers and also influence other managers' norms and teams' culture. Yes, I haven't been coding, but throughout the past years I've been able to create and form a team with a healthy culture that has been successful.
Large tech companies have more than one "culture" which various across departments and projects. When I found myself stuck in an unhealthy project I would look for opportunities to transfer within. Usually I was able to find a team that suited my interests and way of working.
I'm in agreement with you that it's about the business and having the passion and motivation to solve the problems of that business. When I look back on my software career and think about the projects where I didn't have an interest in solving that particular problem, I would always revert back to just learning the technology for the sake of learning. For me, just learning and not applying was never as fulfilling as doing both.
Large tech companies have more than one "culture" which various across departments and projects. When I found myself stuck in an unhealthy project I would look for opportunities to transfer within. Usually I was able to find a team that suited my interests and way of working.