Are there any situations where it is in the author's best interest to seek obscurity?
There is at least one situation where this is likely the best option. If an author has plagiarized another, then deliberately seeking a venue with the most restricted possible access might be the best way to Aim for Obscurity, to minimize the odds of being caught.
To illustrate how easy it is to check for plagiarism in an open access environment, try googling this paragraph from my blogpost Aiming for Obscurity:
In brief, the term refers to authors who do not publish in open access journals or self-archive their works, as well as journals and publishers that are not adjusting to the open access environment.
I just did this, and the blogpost came up first, even though it was just posted earlier today! In case my search was influenced by my previous search behavior from this computer, I would be very interested in hearing what happens if someone else tries a search for this paragraph.