The most offensive part of this whole enterprise is that it is aimed at making life easier for administrators, not better for kids. The social life of childhood is frustrating and unwieldy for educators, so they respond by making childhood less complicated. Indeed, it's worth noting that the psychologists the New York Times spoke to oppose the practice.
In his 1998 book, "Seeing Like a State: How Certain Schemes to Improve the Human Condition Have Failed," Yale political scientist James C. Scott documents how over time states of all kinds -- democracies, monarchies, dictatorships, et al. -- try to make their populations more "legible." What Scott means by this is that for governments to help or control people, they must first organize them in a way convenient to planners. For instance, in many societies, last names are an invention of the state, so governments can distinguish citizens more clearly. Scott documents all sorts of massive state failures that ignored human nature and common sense. The war on best friends strikes me as a perfect retail example of this wholesale line of thinking.
There's a lot of kindling here for a big culture-war conflagration. Many conservatives, myself included, see the building blocks of Brave New World Nanny-Statism (not to mention a perfect example of mission creep in American education).
But we shouldn't get ahead of ourselves. Liberals believe in having best friends, too. And conservatives understand that educators should try to dissuade bullying and blunt the sharp edges of cliques. Administrators are free to complain that best friends make their jobs harder. And we, as a society, should simply respond, "Then your job's harder."