It looks like I can reduce the parts count even further.
Now it's 2 MOSFET, 2 diodes and 1 capacitor.
It looks more like a classic oscillator to me now...
I am tempted to replace the diodes with MOSFETs to increase the efficiency but that would complicate the design since they would have to work when the capacitor is discharged.
Startup is still a problem. I can make it start to oscillate from a low voltage but it relies on transients, to make the coil react.