A former Baltimore Sun editor used to get exercised over the phrasal verb go missing, which he denounced as a stinking British import, and I have encountered other editors of the same stripe.
I think it probably was originally British, gaining familiarity with Americans through all those Inspector Morse episodes on PBS. It is indeed a handy neutral term that has taken root here when it is not clear why the person is absent: wandered off, fled the jurisdiction, kidnapped, abducted and probed by space aliens, whatever.
But I understand the native suspicion of overly enthusiastic Anglophiles. There is a ripe example in Randall Jarrell's portrait of the college president's wife in Pictures From an Institution: "People did not like Mrs. Robbins, Mrs. Robbins did not like people; and neither was sorry. ... She had been a scholar once, and talked somewhat ostentatiously of her work, which she tried to keep up. To judge from her speech, she was compiling a Dictionary of Un-American English; if lifts and trams ever invade the North American continent, Pamela Robbins is the woman to lead them."
Standing sentry for the appearance of those lifts and trams is Ben Yagoda of the University of Delaware, who at Not One-Off Britishisms chronicles infiltrations on these shores. (M Lynne Murphy's Separated by a Common Language blog patiently compares U.S. and British equivalents.)
It is not only here, of course, that observers are on the lookout for invasive species. As surely as the swallows return to Capistrano and the buzzards to Hinckley, Ohio, the British press returns regularly to the subject of odious Americanisms that have infected the Mother Tongue.
Can't we all just get along?
No doubt there is risk of affectation when Amurricans mimic limeys, and the reverse. But it's a big language, with lots of interesting morsels to roll on the tongue. As I pointed out earlier this week, that fully Wodehousian word rannygazoo was originally American slang. And since English is a language that has always greedily and without compunction taken whatever it likes from other languages, there seems little point in denying our own tastes and preferences.