I think people that write articles that start with long anecdotes (which are usually linked back to in the end) are trying to emulate the writing style you'd usually see in newspapers and magazine going back decades. It doesn't work well on the web I feel when there's an abundance of information where you're trying to filter through what's useful to read and what isn't. It's different when you're e.g. reading a magazine and have limited articles to pick from.
It doesn't work in magazines and newspapers either, but certain writerly circles (mostly American ones) consist of telling each other that this kind of fluff is a good idea.
You will find this kind of writing in Time magazine, but not in The Economist. Which is one of many reasons why TE is a better paper.