They work quite well in a physical magazine. They're much larger than the surrounding text, and catch your eye as you flick through the pages. They're chosen to pique your interest so that you end up reading the whole article.
On a website, where you only see these pull quotes if you've already made a decision to read through the article, and they're not even any bigger than the regular text, they make no sense at all.
Thank you for this feedback! I'm the editor at Capital Daily, and I struggled with the decision whether or not to use pull quotes. I agree that our site's pull-quote format is too small, so they look weird, but I did want to give people who are scanning a chance to drop in in different places. I guess that wasn't achieved here.
"a pull quote ... is a key phrase, quotation, or excerpt ... serving to entice readers into the article or to highlight a key topic ... They can add visual interest to text-heavy pages with few images or illustrations"
You have to understand that the average reader doesn’t read the whole article. Most just skim and take in the headlines, subheadings, pull quotes, and pictures.
But, that’s not the stated rationale for a pull quote, which is technically meant just to highlight an important piece of information from the article.