I once bought a 600 watt microwave thinking it should easily run on a 1000 watt genset. Wrong.
As you note, microwaves are rated by power output into the cooking chamber, not power draw from the power line which can be a lot higher.
Fortunately the microwave wasn't very expensive and in the end it only lasted a couple weeks before it crapped the bed and went to the landfill.
This may or may not relate to the failure you experienced.
Generators do not deal nicely with larger loads that are suddenly applied or removed. Some generators are much worse in this regard than others. When a larger load is applied, the engine has to consume more fuel to provide that power. What happens instantly is the output voltage of the generator drops, and the engine slows down. When the engine slows down the governor kicks the throttle wider open and over a second or two the engine speeds back up. If the big electrical current draw was from something like a motor starting, the start up surge (which can be several times the running current) might get done just about as the governor has the throttle wide open trying to get the generator back up to speed. The result of this sudden reduction in electrical load is that the engine revs way up to too high a speed which results in too high an output voltage until the governor can get it slowed back down.
So what this means is the output voltage from a generator tends to sag and surge far more than normal power line voltage when loads turn on and off. Cheaper and/or older and/or generators just barely powerful enough for the loads they are running tend to have a lot worse problem with voltage sags and surges than better, more expensive, and larger generators.
A quick and dirty test is to take a large space heater (or several of them if the generator can handle the load), and plug them in and unplug them. When I got my current used Honda generator, it worked fine powering one space heater, but could not handle 2 or 3 space heaters getting suddenly turned on and off. The generator pumped speed badly. The local small engine place could not fix it, but when I brought it to a Honda Generator sales place service department they found the problem and fixed it. I have had good luck with 4 and 5,000 watt Coleman generators, but have seen all too many inexpensive generators where the output voltage sags and surges were enough to damage equipment.