It doesn't matter if someone could just Google it. The NYT shouldn't make it easier unless there's journalistic value in doing so. Spite doesn't count.
Would you feel the same if this was a story about a sexual assault victim instead of a blogger?
He's a blogger, not an assault victim, but his concerns are similar. Avoiding harassment and physical violence against himself and his family. Avoiding negative consequences on his career.
Why are his concerns less valid than anyone else's? "Because I disagree with him" or "because he talks about ideas I don't think should have a platform" aren't good answers. There's no allegation he hurt anyone, did anything illegal, abused his influence, or acted in bad faith. So what's the journalistic purpose of exposing him?
Should his identity have been withheld because it would hurt his career, or he might be harassed?
It's not like he was trying to hide from the NSA or anything. The goal was "I don't want my patients to see my blog on the first page of Google results when they search for my real name". The NYT completely obliterated that.
Additionally, the fact that the NYT chose to smear him means that now many potential patients may google his name, see the hit piece and nothing else, and decide "yikes, better not go there".
The article wasn't flattering. It wasn't a smear or a hit piece.
It's not a requirement to offer this protection, but declining when it's asked for by the subject of a story is certainly a weighty ethical decision that respectable journalists don't take lightly.
Outing someone in a vulnerable position simply because the journalist disagrees with their ideas and wants to harm them (vs. deciding it's in the public interest) would be a very clear breach of ethics.