Why is a power amplifier most efficient when operating close to saturation?
It makes no difference whether it is a power amplifier or a DC supply feeding a resistive load via a variable resistor. The variable resistor is, in effect, your amplifier. And, as engineers, we learn that when the variable resistor has the same resistance as the resistive load, maximum power is wasted in the variable resistor. See the maximum power transfer theorem.
But we also know that when the variable resistor is set to close to zero ohms, maximum power is transferred to the load resistor hence, efficiency of power transfer is also the greatest.