If there was ever a sector where the Chinese ability to cost effectively churn out good enough then drone applications are it.
Getting a handle on how nations are doing for jet powered drone is a bit tricky because of limited data available for what is primarily a military application. But, if we take model aircraft applications as proxy, then my understanding is that the Chinese company Swiwin (est. 2013) is already undercutting everyone else.
They actually do. The segment of small jet-powered (military) drones is rapidly growing and isn't served by the big jet engine manufacturers.