Ideally. Or in case they might agree with him. The therapeutic relationship should be one where the therapist offers the patient a mirror/lens/toolbox regarding their own views and behavior, and a patient's beliefs about their therapist's biases one way or another can interfere with that.

The majority of the psychiatric community disagrees with you. They believe the patient should not form personal relationships with psychiatrists and therapists, including not being aware of a psychiatrist’s positions on political and social issues. They have patients with a wide range of views, some of whom rely on their treatment to maintain their psychological well-being and mental health. It can be dangerous for them to have that relationship disrupted by disagreements about orthogonal issues. Psychiatrists are not primarily concerned to protect themselves, although that is a factor; they are largely motivated by a desire to avoid endangering their patient by disrupting their treatment.

You might want to argue that it’s Alexander’s fault for writing a semi-pseudonymous blog in the first place, but that doesn’t help his patients. And I might agree with the view that exposing him is justified if the article actually revealed something of public interest-that he was a Nazi or a pedophile, for example-but it doesn’t. It’s a thin tissue of spurious guilt by association and mischaracterization. The NYT have decided that any risk to Alexander’s patients is less important than naming him in a low-quality hit piece. What they have done is immoral.

I think the idea is that therapy should focus on the patient and anything that might distract the course of therapy is harmful to the patient (deprives them of effective therapy treatment).

Seems incredibly plausible to me. Not that his patients would disagree with him, but that they would no longer trust him.