Dead time between left and right isn't the problem. Dead time between top and bottom is the problem. As shown, the top and bottom gate voltages are (more or less) equal, so any time the gate voltage is above Vgs(th) for the N channel and below Vgs(th) for the P, you're shorting out the supply.
And with op-amps generating the waves, not only will the crossover time be relatively long, the saturation voltage may not even exceed Vgs(th) for either transistor, so it never fully switches at all, it just sits there smoldering.

Tim