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The secondary winding should be operated in flyback mode only. So get rid of half the parts, use a single diode and filter cap and make sure the windings are properly connected for flyback operation. If you keep the upper diode, then windig should be reversed.
If the voltage is too low, change the turns ratio, instead of using the two diodes.
What they've got is a PFC, that is a boost stage. But the principle is similar.
Anyway, using the transformer that way means you do not have any real way to limit the current in the diode that conducts during the transistor on time. You would rely on the transformer's secondary impedance only, which may not be enough. That is why I suggested you use it in flyback mode only.
In flyback mode, you have a much better line regulation. Because you rely on the voltage that is across the LED's for the output voltage (and the turn ratio of course). In forward mode, you need to calculate with the difference between the input voltage and the LED-voltage.
In forward mode, you can have also big transients on the auxillary voltage at startup, because the LED-voltage ( and capacitor voltage) is zero. With the flyback, the auxillary voltage will ramp up toghether with your LED-voltage.
To my opinion, the circuit is neither wrong nor right. It has specific properties. The PFC datasheet tells, why they preferred this circuit: cause it's output is proportional to output bus voltage, in other words, almost constant. A flyback circuit won't work in this situation, respectively need a large filter cap. In your original buck variant, the voltage is proportional to input voltage. This may be either wanted or unwanted, I don't know.
The said problems of inrush current can be handled, if necessary.