The position of Scott Alexander seems to be very, very different - entirely opposite from "patients of Alexander might have a legitimate interest in knowing about his writing." the general practice seems to be that it's imperative that patients do not know about his writing, as that affects the therapy in a negative manner. If a potential patient reads Scott Alexander's personal blog and knows who he is, then Scott would be ethically prohibited from taking them on, they are not a potential patient any more; if a current patient does the same, they may have to terminate the relationship and transfer to a different therapist solely because of that. It was also clearly stated that patients reading his public blog and knowing who he is is incompatible with the employment at his previous clinic - if he'd be outed, he'd be fired and all the patients would suddenly have to find a different therapist, which isn't good for the stability of their treatment.

A therapist is supposed to be a blank slate to project what is needed for therapy. The patient is not supposed to know whether the therapist is married or not or divorced, because that may affect (not in a good way) how the patient handles revealing their relationship trauma - the therapist should either appear neutral (to freely allow the patient to project whatever they want to project) or be free to create whatever impression about himself works best for the therapy. It would be near-taboo for a therapist to have a family picture in the room visible to the patient, and in the exact same manner the patient reading about therapist's personal life can only hurt therapy.

The patient should not know the therapists political opinions or moral opinions about various ethical dilemmas, as that may hurt the therapy as it involves the patients' current personal stresses. One patient might need the therapist to show agreement to their viewpoint in order to open up more, another patient might need the therapist to confront and dispute the same viewpoint. The therapist must not be taking sides based on their personal beliefs, they should project whatever viewpoint is needed for the particular patient at this moment. For this reason, various symbols of political or advocacy affiliation are near-taboo, all the therapist profiles are explicitly neutral, and (again) a patient reading their therapists personal blog is bad for the effectiveness of their therapy. A patient isn't supposed to choose a therapist based on their political opinions anyway, if the therapist lets out their politics in therapy, then they have failed at their job and the patient should get a better therapist no matter if they liked those politics or not.

Patients have a legitimate interest in knowing the professional qualifications and professional reputation of the therapist, but all the personal life and personal opinions of the therapist are out of bounds as it disturbs the relationship with the therapist - the patient has a right to read that, but it would force an ethical doctor to "fire" that patient. As far as I understand, currently Scott Alexander has to filter potential patients based on whether they read his blog as it's not anonymous anymore, but that's manageable as he now has a separate practice. But if indeed all his patients would have a "legitimate interest" in reading that and would actually do so, then he would have to stop blogging about pretty much everything meaningful because that would be incompatible with treating patients properly.