The circuit used allows it to work using only a single rectifier diode. The cap in series with the transformer insures that there is no DC flowing in the transformer winding, and that all of the transformer winding is utilized for both halves of the cycle.
To do this without the doubler, you’d need either a full wave bridge, or a center tapped transformer winding. The full wave bridge rectifier requires four diodes, and the center-tapped two. The center tapped version “wastes” half the copper, as only one half is used for each half cycle.(and you’d need four times as many turns)
You also get to ground one end of the transformer secondary, which is a good safety feature.
Also, the doubler configuration used is also known as a “charge pump” in other applications. The capacitor transfers a limited amount of charge each cycle, so acts as a non-dissipating current limiting device. This along with high leakage reactance in the transformer gives a fairly constant current drive to the magnetron, so you get pretty steady output (high utilization of the magnetron) without needing huge, heavy, and expensive filtering elements in the power supply.
But the main thing is making the transformer cheaper. A first order analysis indicates you need about the same amount of copper. (twice as many turns, but half the cross sectional area)
But practice is not so nice. Due to the higher voltage,the insulation has to be thicker, (by twice) and that takes up more room, and that means you need more copper, because the insulation takes up room, so the winding as a whole gets bigger, and the insulation blocks heat flow, So your wire actually has to be more than half the cross sectional area so that it has less resistance and generates less heat to have trouble getting out, and that fatter wire also makes the winding bigger, which means it has to be longer, which means it has to be even a little more fat.
The circuit has evolved over the years, and is actually fairly elegant: Cheap, simple, durable, and easy to trouble-shoot. (with the right training…poking around in a hot microwave oven is a good way for the untrained to become unliving!)