The minimum wage floor gives the job seeker some bargaining power, in a sense.
you may be missing the point that it is in the interest of some workers to work for a wage that is not high enough to sustain themselves. Imagine a teenager who has all of his needs met by his parents and gains valuable work experience from the low wage job.
More aged workers who need a higher wage to sustain themselves would have to differentiate themselves from these teenage workers in order to compete. Experience would be the large factor here, perhaps with ability to work a full-time, non-flexible schedule being a close second. Those are valuable traits to many companies wishing to hire, traits for which they very well may be willing to pay a significant amount higher to attract non-teenagers to the positions.