I'm genuinely curious because I feel like there's something I don't understand here. What's the link between the club's identity and the way the manager wants his team to play? If the game is about glory, then Ardiles should have been made manager for life!
When you appoint someone like Mourinho (or Graham or Allegri), you know football aesthetics won't be their number one priority. Nevertheless, their teams have a distinct identity or we wouldn't be able to say that. I'll give you that we seem to be the exception to that rule but that's because Mourinho's been trying everything he can to turn things around. Having said that, I expected him to change our players' mindset and make them much more ruthless. That hasn't happened and that's why I'd be inclined (personally) to cut the experience short at the end of the season. Regardless, I'm feel like I'm not getting your point about identity.