For most acts, they do have the other lights on occasionally, although some bands just have red during their whole act, so waiting for other lights isn't an option. Just the slightest amount of blue or yellow makes a big difference in the quality I can get out of the images. I was more curious from a technical standpoint why red is such a difficult color. Pure blue or pure green gives great, stunning results, but pure red doesn't.
On the T2i, solid red light generally tricked the camera into using one higher ISO setting than it should have, and I learned to set it to ISO 800 manually, but the 5D Mark III doesn't have that problem. I use Shutter priority and 1/160 second most of the time and the camera usually always "does the right thing". Bands are moving a lot and 1/160 will freeze the people while allowing the drumsticks to have some blur on them. For facial close-ups, the lens is usually wide open and the shallow depth of field works fine, with the person's face in focus and everything else blurred.
Forgot to mention -- I do shoot RAW and it does give a lot of opportunity for making better images afterward.