It seems quite sensible to me. There really is a significant portion of the population that has never even heard of Twitter (remember, there are lots of people who try to sign in to Facebook by typing "facebook login" into Google and blindly clicking on the first result), and they're probably disproportionately well-represented in the Times' readership. It's not about hating the word "tweet," just about "tweet" not being common among the 6.5 billion people who don't use Twitter.
Clarity. That’s the idea. Don’t use words many of your readers won’t understand (except for special effect).
“She wrote on Twitter …” is much clearer then “She tweeted …” and it seems obvious to me that whenever writing formally, you should use the first. Just as you would write “He searched the web …”, not “He googled …” in formal writing.
It definitely has something to do with formal editorial usage than drinking the Kool Aid amongst many institutions. But because of this single word "tweet", almost everyone in pop culture can get a good sense of what this writer is talking about by remembering this new phenom. The only culture barrier that I see is if institutional writers do not want to use the word, then it won't change the Twitter ecosystem.
As a matter of fact, Jack and the team initially did not like using that word. It was only when the Twitter ecosystem starting vastly adopting it that they finally adopted the term. Pop culture, once again, wins.
