That's a straw-man. The desire to stop illegal immigration isn't about "supply-and-demand" any more than the desire to stop the flow of illegal drugs is.
In either case it's about the EFFECTS of the illegal traffic. Of course the U.S. Chamber of Commerce is going to support "comprehensive" immigration legislation. They just want the cheap labor, and they don't mind if all the rest of us pay the price for it.
I, for instance, never gripe about "jobs going overseas" because that's JUST business. Illegal immigration isn't about "just business" it is about an insupportable burden on legitimate citizens and their taxes.