If all of the coils have continuity, then I would suspect the transistors.
First thing is that you seem to expect DC out of your transformer secondary when you put DC to the primary.
Transformers don't work that way.
Only a changing voltage on the primary will cause a voltage to appear on the secondary.
You connected the transformer to a DC powersupply, and got 0VDC on the secondary. This is correct. But, that was a bad idea. Connected that way, a lot of current will flow through the coil, possibly overheating - that can either melt the insulation and cause short circuits in the coil, or cause the wire itself to burn through.
Q9 is supposed to be driven by a pulsed signal. I expect DMMP should be a series of pulses rather than a simple DC signal.
The circuitry around D7 and ZD1 (together with the lower coil and Q10) might form some kind of oscillator (in which case DMMP would be a simple on/off control.)
In any case, Q10 would be my first suspect. Your diagram says you measured 5V on the collector of Q10. That shouldn't be. The voltage there should be fluctuating - that's the changing voltage that drives the transformer. It should measure less than 5VDC if the circuit is operating (your voltmeter can't measure the fast changes, so you will get a sort of average of the high and low values of the pulses.)
Check Q10. Use a diode test from base to collector and base to emitter. Put the red lead on the base. Black lead to collector and emitter. Both should read about 0.7V. If not, then Q10 is toast.
Q9 might be bad, check it the same way as Q10.
But, I suspect Q10. It is driving a coil, and has no protection against the voltage spikes that can occur when switching current through a coil. That transistor can switch around 1A. That amount of current through that coil can cause a large voltage spike if switched off suddenly.