Anything can be the cause of an old transformer to go bad, from weakness due to manufacturing process, to a gradual erosion of the connections over time. The transformer will be the most expensive component you'll have to replace. As long as you're certain the capacitor and new diode are good, and there is no age or heat-related damage to the magnitron, after replacing the transformer, you should be good. You'll be the ultimate judge as to whether the fix is worth the expense. Look very closely at the magnets in the magnitron, though, and make sure there are no cracks in them. They should be visible as you look, between the cooling fins, with a good light.

Magnetrons fail in several ways. The filament can open or short to ground. The magnetic field can scatter due to broken magnets. The filter can break. All of these will result in a no-heat condition.