Why do microwave ovens use magnetrons?

Magnetrons are cheap, reliable, pretty efficient (65% or so- and they tolerate high temperatures so heat sinking is easy) and made with mature technology. They are also reasonably tolerant of VSWR issues (if the user does not put a proper load in the oven, for example). They don't really allow the frequency to change much without expensive mechanical tuning which is not available on consumer ovens- so standing waves tend to appear in the oven.

To get 1000W-ish of microwave power any other way would be more expensive and possibly more fragile. It's possible today, but too expensive. Of course the semiconductor makers are always looking for the next big market, but the oven market is going to have to wait more years I think. One of the few advantages they might have is to allow the frequency to be modulated which could reduce or eliminate the need for turntables and stirrers. However that could have implications in other areas of the oven design such as the door, which is designed to attenuate one particular frequency.

The domestic microwave oven needs high power to cook the meal and high frequency to excite the water molecules. What is not needed is high stability because the water energy absorption spectrum is broad. (1, 2) The magnetron does this cheaply. The low price and low duty cycle of the domestic microwave means that they should last for many years despite falling magnetron output.