Tdelahanty, what you are describing is known as 'class B' operation. It is also possible to have a class A push-pull amp, where both tubes are always conducting throughout the entire signal, even at full power.
This makes for a better sounding amp, and is still more efficent than single-ended. The only way to avoid the degradation of the phase splitter is to build it into the voltage amplifier, IOW make the voltage amplifier fully balanced.