You are missing the point, find me a smash mouth team that can’t block. Passing to set up the run sounds great, find me a team that can’t pass block that is carving out their place in the league with that identity.
Teams certainly want to do things, but without the right pieces they won’t get done. And announcers will have the cliches of “team X is just trying to find their way” or “team Y is trying to get back on track”.
Strategy is a choice, so-called identity is the result.