The quote is misleading since there is not "a" chef in the kitchen. There are many different chefs, i.e. pantry chef, soup chef, fish chef etc. Wikipedia lists many others.
What the quote is thinking of is the "chef de cuisine", which is the overall manager of the kitchen, the CEO if you will. There is also the whole executive hierarchy we know from our IT companies, e.g. sous-chef (second in command), commis chefs (department manager) etc.
So, your fullstack dev is first and foremost a someone who can cook well. But even if you cook the greatest delicacies, that does not make you a good manager. On the contrary, there are people who are actually trained to be managers/leaders. The skillset, and daily work, of managing the logistics of a large kitchen is vastly different from the cooking itself.
Instead, a fullstack dev specializing in AWS, Azure, CI/CD, "xxx as code" or any of the other tools that we associate with DevOps, would be comparable to a soup chef specializing in soup or a fish chef specializing in fish (with no managerial tasks attached to the job description).
And finally, remember that in IT, "Agile" is all about removing leadership from the immediate development process and placing leadership where it can do good: enabling the devs to do their work. For that, they do not need to know that much about software development, or be a great dev themselves.