This would be the case in a normal free market, but unfortunately the labor market is not a free market as people don't have the choice to work or not to work - they need to work to survive.
As long as millions of people are out of a job, there will be millions desperate to work even for under minimum wage, ready to replace anyone who quits.
This is why there are two things needed - first, a minimum wage not defined in dollars (to account for rent and other CoL differences) but in "is enough to rent a decent home in the area and live without a second (or worse, third) job or government assistance programs", and second, a universal basic income that fulfills the same requirement. Only then, when people have the freedom of choice between working or not, will labor prices reflect the true value of the labor.