Does A Low Minimum Wage CUT Demand At Min/Wg Businesses?!

Late thought on the minimum wage which shows how easy it could be to pay poor Americans much more:
if hiking the minimum wage redirects demand toward businesses that pay the minimum wage because they tend to serve a lower income clientele: does that mean that NOT hiking the minimum wage to keep up with inflation -- or growing average income -- would actually tend to CUT demand (over time) at minimum wage businesses?

Even without the "redirecting demand to their own businesses" phenom', what would actually take place on that old eco 101 price/demand chart when the min/wg is raised is: the same thing that would have happened without the wage hike: the demand curve would keep inching up (nominally) in step with inflation raises to employees across the economy -- while the price curve crept up respectfully behind, afraid to alienate consumers.

American workers have got to shake loose from the Newt Gingriches and their homeopathic (as in unscientific and always proved wrong by experience) unfettered-market theories.