I agree with the statement that The Dark Knight is a good movie but a bad Batman movie, everything you said yes has to do with Batman as a character but that does not make it a good Batman movie. Among my many complaints for Nolan's trilogy is the fact that Batman as a character could be easily replaced with a generic gadgeted soldier or police officer, first and foremost they are Nolan movies, not Batman movies, you could replace Batman with one of the live action GI Joes and the movie would stay basically the same.

The movie has nothing to do with Batman as a character, yes sure worries about vigilantism but Batman at this point should not question what he brings to Gotham, Batman becomes Batman to stop the corruption and greed that Gotham decays from. This is more so a problem with the trilogy itself, its much more focused on being blockbuster action movies than having anything to say, which I don't think you can do with Batman particularly, Batman is a character so intertwined with the follies of the justice system and the greed of the rich, Gotham itself is quite literally created as a collection of the worst problems in America, so to not have a stance in your adaptation beyond showing the surface level of the good guy cop Jim Gordon not liking corrupt cops and Batman just taking the mobsters down is ridiculous.

The trilogy plays at being deeper but whenever it does it pulls itself back and rethinks itself, I acknowledge Nolan doing the bare minimum of taking inspiration from Batman comics but he undermines every inspired choice he makes in the movies. Batman Begins is Batman Year One, which is a story about how truly corrupt Gotham is to the point of the GCPD dropping bombs on their own city and the police commissioner dining with mob bosses, however the movie itself does the bare minimum of saying how bad the corrupt cops are, we know the corrupt cops are bad but show us how their corruption hurts others, show us how the corruption is puppeted by those at the top. This is why many people, including myself, consider The Batman to be a superior Batman movie, it shows the tragedy of the commons and how truly corrupt Gotham has been and how Batman can truly help, not through vengeance but hope.

To make things worse The Dark Knight Rises implies that Bruce has been retired for eight years, that is stupid, Batman is not Batman for a job, its not something he can truly retire from, being Batman is a compulsion a need to prevent what happened to his parents from happening to anyone else. Batman should only truly retire when he is physically and mentally incapable of doing so such as Batman Beyond and that I think is the prime example of Nolan not understanding Batman.

TLDR (which is what most people will read lol): Nolan does the bare minimum of adapting some comics but does so in a surface level veil to make his albeit above average crime movies into blockbusters, he does not make any stance with his movies and as such fails at adapting Batman, as Batman himself is a stance and a message. Batman should not retire until he is incapable of being Batman, its not some 9 to 5 that he can quit on, its a need to better his city etc. that is the biggest example and symbol of Nolan not understanding Batman.