This series capacitor is not practical as lamp current is low frequency (~150Hz) compared to half bridge PC power supply and the capacitance has to be rather high. And is actually not necessary, as the midpoint of electrolytics serves exactly the necessary purpose, with infinite impedance at DC.
In electronic ballasts, there is typically a sense resistor in the source of lower MOSFET(s) of half-bridge/bridge, so ballast controller can see lamp current in both half-periods, and regulate/balance it.
Also, in a classic forward-type push-pull PC supply, power transformer does not serve energy storage function and so has no air gap, and has large inductance, so can saturate easily. Energy is stored in the separate powder cored inductor after rectifier diodes. In HID ballast, the inductor serves the energy storage purpose, is air gapped, and will not saturate, as it has to sustain low frequency component of lamp current.
