I hate pull quotes. I especially hate them for online articles. And especially when they are on phones. But given the single-column-default of most online content currently, they are bad on any screen size.
Jeremy Keith already wrote about this, and his post includes examples showing why they are bad. You can probably read it instead of the following few paragraphs since the sentiment is the same and his writing style is better. It still didn't tell me why they exist though..
In magazines or print media I can understand why they might make sense - someone flipping through pages and skimming the content sees an interesting quote in big colorful text. That draws you in to read the article, and you have an easier time skipping past that bit of text. Even so, I usually get at least halfway through a sentence before I realise it's the same damn one as 3cm away, and someone thought it was a good idea to write it twice..
But for online articles it makes no sense to me. In an online article, I am not casually flipping. I don't need an attention-grabber 3/4 of the way down the page to grab my interest. I am trying to read the piece!
You usually can't skip over them - they are eye-catching and large enough that you always see them. So you start reading the sentence and realise that it is the same as the one 3 lines above!
I don't know if there is a good reason for them being there, or it's still people trying to imitate print. Maybe this is for people who don't like to read the text, but still like to say they scrolled to the bottom of the page. Skim a few key quotes and call it read.
Maybe they'll make more sense with display: grid or other clever layouts - I'd be fine with pull quotes that were off to the side of the page (maybe). But full-width giant text taking up my whole phone screen? I do not want this.