Why use RMS instead of the average voltage of half the cycle for power
They're not the same, in general. For a sinusoidal waveform, the difference is about 11%. The square root of the average of the square of a function is not the same as the average of a function. 



The average over a half cycle is 2Vp/pi = 0.6366 Vp

The RMS is Vp/sqrt(2) = 0.707107 Vp



The ratio is 1.11. 

The ratio will be different for different waveforms, but in general the RMS will always be higher. RMS gives you the power transfer into a resistive load, average does not (except for the degenerate case of DC). 

In some cases you may actually want the average (but not for power into a resistive load). 

Cheap multimeters often just measure the average of the rectified waveform and read 11.1% high, assuming a sinusoidal waveform.