It is hard to make a 555 put out a 50-50% duty cycle. You have to include a diode in the timing network to make the charge time and discharge time symmetrical.
Also, what is the logic inverter? If it is CMOS, then the FET gate driven from the 555 has much better drive than the gate driven from the inverter. You could add a second inverter to make both gate drives equally bad...
The right way to do this would be to get a transformer with a third winding (also center-tapped), and use that to drive the gates. That way, the period of the oscillation is determined by the saturation of the transformer core; not by a 555 which knows nothing about what is going on inside the transformer core... The way you are doing it, how do you know that the period of the 555 is right to drive the transformer?