Reducing standing waves with phase randomization
Assuming that you are not changing the phase too much, you are basically turning sinusoids to narrow-band noise. If there is a room frequency response node that strongly attenuates a sinusoid, the total power of narrow-band noise centered at the same frequency will not be attenuated as much. It will survive with a dip in its spectrum. Room modes or frequency response peaks are usually not as severe and narrow as frequency response nodes.


An example speaker and room magnitude frequency response (CC-BY-SA 3.0 by Gregory Maxwell)

However, I wonder if your application affords the degradation of audio quality due to the modulation. Also, the room frequency response is not affected so if the test audio is noise, you won't get any sort of improvement.