I imagine they're used for SEO now as search engines may weight that text more heavily. This is achieving the same end goal you pointed out of the original pull quotes.
I doubt it has anything to do with SEO. Some crawlers might try to extract section headings from content based on font size, but in this case the pull quote may even confuse this.
Vanity Fair didn’t want to optimize search rankings based on “The crime boss was ‘A CRIMINAL STAR in Romania,’ going by a variety of nicknames, among them ‘PIG HEAD’ and ‘GODFATHER.’” My bet is that they long ago told the web design team to make a theme that captures the feeling of their print magazine. The designers thoughtlessly included pull quotes in this theme. The CMS was adapted to allow insertion of said quotes. The person who prepared this article for publication added them hastily, scrolled 30% of the way down, picked two sentences that are near each other and are both about the same thing (Romanian criminals, not rare books), and moved on to the next assignment.
