You have to distinguish between deleting and closing here:
Closing mainly prevents users to add further answers to the question. This is the method of choice for questions that we consider a bad fit for the Stack Exchange format or this community. Whether this applies here is indeed debatable, but this is another discussion – because closing does not make a post less visible and thus does not solve any possible privacy issues.
Deleting is intended for posts that have no lasting value whatsoever or are causing other severe problems. This would indeed make the post considerably less visible. Note that on Stack Exchange deleting is not the method of choice to deal with posts where the author revealed too much. As it stands, the answers may still help somebody in a similar situation and therefore I see no reason to delete the question.
In this particular case, we (i.e., some moderators) redacted the question’s (and some answer’s) revisions to remove a considerable amount of irrelevant but identifying information and it was disassociated from the asker’s account.
There shouldn’t be any publicly available information that allows to identify the asker anymore (if we missed something, please flag it).
There are probably thousands of people who could be the author.
All this happened almost four months ago.
We haven’t heard of the asker since then, so he either was satisfied or escalated the issue to Stack Exchange without success.
As a final note, given that this question has been up for almost four months now, if it has done any damage to the asker, it already did, most likely before the asker even wished to delete it.