Yes, it's gate bias that's critical to maintain, indeed, dropping it unexpectedly will most likely destroy the transistors due to flyback of the supply inductor!
I used the same mechanic here, varying the supply voltage with a linear supply (a switching supply would also do well, I just didn't need the efficiency).
Behavior is kind of weird at low voltages and light loads, as it oscillates in weird modes, or squeggs, or quenches entirely. It also can't go down forever, as the gate bias itself is a power source, and if it's sufficient to cause oscillation, it'll generate some output (in this case, something like 200V). Such is the consequence of such a simple circuit.
Mind that you'll similarly need a nice high voltage resistor, to make the sense divider.
Also, make sure you're using the right kind of flyback transformer, without internal rectifiers. You can't run a C-W from the DC kind, at least not in any reasonable way.
Tim