RMS power and phase angle control
For a resistive load, common sense tells us that for a symmetrical signal, 50% conduction will give 50% power. But since power also equals Vrms^2/R, the Vrms must be 70.7% at 50% conduction. 


  Assuming a duty cycle of 50%, this gives 50% current and 50%
  voltage...


So, this assumption is wrong, RMS voltage is 70.7% and the RMS current is also 70.7% at 50% conduction. 70.7% * 70.7% = 50% power.

The following graph was created using instantaneous voltage and current over every 1/2 degree.



Charles described most everything else well, I will only disagree with half of one paragraph.


  If those assumptions were accurate, multiplying RMS current by RMS
  voltage on either side of the controller would provide the correct
  result for power and the two would differ only by the power dissipated
  in the controller.


I don't think that this will apply for the input side of the controller. At any instant when the current is zero, the voltage can't contribute to the power.