But, be careful not to read what he writes out of context. The two points before this statement are:
The first is that, like a career in science, a PhD is not for everyone. It requires a peculiar mix of intelligence, discipline, creativity, rationalism, stubbornness – and sheer nerdiness. Different people have these in different measure, but a successful PhD student has a healthy dose of all.
The second is that a PhD is hard. It’s meant to be hard, not because inflicting pain is necessarily fun, nor because some scientists are ‘dementors’ (see this interesting post by Zuska on that subject), and not because your PhD is expected to solve the mysteries of the universe. It’s hard because it is an apprenticeship in science: a frustrating, triumphant, exhausting, and ultimately Darwinian career that will require everything you can muster.
In the context of "a PhD is hard" and having the right mix of "intelligence, discipline, creativity, rationalism, stubbornness" it is probably true that "almost everyone who starts a PhD and sticks around long enough ends up getting one". But, sticking around means passing classes, passing qualifiers, and passing numerous other barriers that are put there to ensure that you have the ability to finish your PhD while maintaining the quality standards of the university.
Anecdotally I know of very few people who stuck and around didn't get their PhD in the end. But, I do know of exceptions - people who stuck around and tried to get their PhD but failed.
This article from the Chronicle of Higher Education from 2013 claims that only 50% of students finish their PhD, and gives an anecdote that only 10% of one class finished.
So, in the context of what Chris Chambers writes it's more or less true, but in the broader context of people doing PhD's it's probably not.