I've lost track of the rationale you're trying to give me for why the NYT is obligated to pretend Alexander's identity wasn't public, or why the NYT is somehow a party to Alexander's professional practice.
It's not _obligated_ to, no one says it's illegal. NYT can decide to be an asshole and reveal it despite the request not to. Which it has.
Is there a real argument for why the NYT _should_ publish Scott's name? Other than "it doesn't care whether that hurts the subject, and it expects to get more clicks this way".
Not really. The NYT was perfectly willing to protect the anonymity of a different psychiatrist blogger. Yes, this person did a better job of keeping their identity a secret, but the justification was largely the same.
And let's not pretend that journalism is only about sharing information. There's a great deal that journalists don't share (or even try to actively suppress) for many different reasons. Partly because there are a lot of ethical considerations when publicizing information.
Worth noting that part of the issue is that, with psychiatry and counseling in particular, it really doesn't matter what he was writing about or what his politics are. Any such publicity compromises the blank slate that counselors are supposed to present to their clients. While this does mean that mental health professionals have an obligation to take care with their public image, it's not unreasonable to ask others (including journalists) to respect the particular cost they face with any kind of exposure.