It looks like deleting a site from one's profile no longer works.
I've updated an answer I provided previously to reflect the latest site layout however, as per my comment (at bottom of the question) the functionality seems to be broken.
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Deleting a site from the "Your Communities" section of the site switcher has nothing to do with deleting user profiles. :)
That section exists to allow people to have a way to access their frequently used sites or just sites they're curious about faster.
I updated your answer on the other post with current screenshots/instructions. Sorry about the confusion here.