If onboard starters for DF were such a great idea, we would all be using them. Same for prop planes. Too much extra weight and complexity for too little gain.
With turbines, it's a little different, as with electric start, the whole startup sequence, which is much more complex than that of DF, can be controlled by the ECU. Besides, it only takes a very small motor to spin a turbine, as opposed to a DF, and the plane has to carry a battery pack anyway for the fuel pump.

I know, some will say airstart is superior anyway, let's not go there!

Anyway, my point is, I think that onboard starting is a blind alley, technology-wise.