Very. The doi is now the unique key to identify a scientific article.
Researchers like it because it is an easy way to get canonical metadata without having to go through Google Scholar or (dog forbid) something clunky like Science Direct. You just put the doi and bang, you’ve got the article. No issue with a typo in the volume number, or authors who can go by different names (or the same name written differently), or anything like that. It has make managing bibliography databases massively easier.
Editors like it because it’s a very efficient way of checking the information in the bibliographies of submitted manuscripts.
I have no clue about their examples, but in the real life I have never had a link rot issue in ~10 years handling thousands of references.
First, I don’t know how popular it is.
Second, look at what an absolutely poor job the doi.org site themselves have done keeping their showcased example links from their demonstration document from breaking over the years. See my comment elsewhere here about what I found on archive.org. What am I missing here? It looks like their link rot is close to 100%.