They don't need DC to operate, as the tube ia actually a diode, which just so happens to have a load built into it at the anode. It is quite happy with AC applied to it.
Same with a microwave magnetron, the capacitor and diode is just there so that the transformer can be cheap, as it does not then need to be wound for 2kV isolation on the coil and the former, which would make it more than double the size and also require it to have a big core gap to handle the saturation, or it will need a big ferrite bias magnet to keep it in the linear core region.
This one is oil cooled and has a very low duty cycle so the heat dissipation is not an issue for it, they just went for massive overkill as it worked out cheaper on a low production run unit. These tubes generally are run for under 100 hours total at 4-6 seconds at a time, not like a microwave where it will do thousands of hours at full power