I think this isn't quite what "privilege" means, at least these days. People talk about "white privilege" for example, meaning that people who are white can do XYZ or avoid ABC, unlike other people.

In the example the author writes about, the privilege is not "being a bag grammar person", it's being a high-ranking person. The bad grammar is the thing that those people are able to get away with.

IMO, he's confusing the disease with the symptom, so to speak.

Separately, I would say that high-ranking people can definitely get away with short emails, and to some extent brusque emails. Bad grammar is perhaps just the next domino to topple.