Gate supply needs to be from a separate input. If gate voltage falls, both transistors turn off, and one of them carries the supply inductor's current in avalanche. Poof.
This circuit is 180 degrees completely flipped from most SMPS. Where usually you want both transistors off during switching, here they must both be on. As well, the idle "off" state has them both on (so the output voltage difference is zero: just as the output current flow is zero, in a half-bridge inverter with both devices off). Thus, the default gate voltage must be high.
(By the way, the supply inductor is emulating a current source. When you short out a current source, its power output is zero. An inductor is not a CCS at DC, so this obviously would be a bit of a problem. With a constant voltage supply, you must reduce that to zero instead. While keeping the gates high. Then once the energy decays, the gates can be turned off.)
Tim