Am I missing something obvious? I take pictures which are nearly always 3:2 or perhaps 4:3, so why should I want them to be stretched to fit 16:9?

It's because that's the cheapest aspect ratio of LCDs, with the economy of scale of producing for everything from mini laptops to portable DVD players. It used to be the case that only the very cheapest photo frames were 16:9, but they've now sufficiently conditioned us into believing "wide-screen is best for everything" that they can get away with it on more expensive models too.