Twee is pervasive, genteel and hard to bear, pixie-haired, wide-eyed and precocious. Twee is also out of hand, and more complicated than it seems. See, though being twee is often regarded as a negative quality, tweeness is not necessarily insufferable.
Obviously, he's into language in a general sense because he's a professional writer and seems to be doing well (and lucky to be working in the shrinking newspaper biz), but he *really* seems to be into language because he shapes his piece around the word "twee" to the point that I wonder if his intention was to write about the word or about pop culture (which seems to be his beat). It's almost nerdy, which is refreshing to see in the simplifying media world. (I'm a proponent of clear, simple writing, so it's not a knock against what 21st century mass writing has become, just an observation.)
But back to the American vs British usage of the word. Because I pretty much never hear people say it in the USA but have heard Brits use it, I assume it's not at the top of people's minds here. So it's surprising that he shapes the essay around the word, as if people have heard it often and are nodding their heads in agreement. Are people sick of the word, or concept? I don't know if they hear it enough to get sick of it, or even know what it really means and how it can be applied.
I'm not saying what he's doing is wrong, it's just atypical because his post seems like it's meant to be a review of some TV shows, but it's also a review of American culture, yet also expresses a fascination with the word itself. His enthusiasm is obvious, and his writing seems to be really good (which is why he's living the dream).