I don't think anyone believes that wages are being artificially held down by some kind of employer cartel or something. The problems are elsewhere. First, with a pure supply and demand situation a very tiny oversupply gives all the power to the demand side. Imagine a theoretical relatively frictionless economy where there was a supply of 110 workers, all of whom needed a job so their families didn't live in poverty, and 100 jobs. Any time a worker tries to threaten with "I'm going to quit rather than work at this wage", the employer has 9 extra people unwillingly living in poverty who are happy to come work for them at that wage and everyone knows it. As a result, absent organized labor, which is no longer much of a force in the U.S., the employer can drive down wages pretty far compared to the situation when there were 100 workers and 100 jobs (again abstracting away friction and churn and other complexities to make the underlying point).
Even worse, the impact isn't symmetrical when you have a supply shortage. The impact to a company that employed 20 of those employees and could only manage to hire 19 people is probably a reduction in profit but only by say, 1/20th, or maybe more but generally the pain is spread out over a base of some kind. On the other hand, the worker who isn't working loses 100% of their income, they're proportionally impacted far more and the pressure for them to take any deal they can get before they starve is far higher than the impact on the employer to take any employee they can get to get from 19/20 to 20/20 strength. Additionally many companies are in a far better financial position to outlast employee demands than individuals are. If you are a 18 year old who just joined the workforce after your parents kicked you to the curb you don't have access to credit or savings to hold out for any period of time without a job. Most companies will have the cash or access to credit to go for quite a while moderately understaffed. Many could go indefinitely, they just wouldn't have as high a profit as they wanted. Again obviously things like unemployment insurance ameliorate this somewhat, but inherently the low, unskilled end of the employment spectrum is an asymmetric environment where employers have more power than employees.
Not to say I disagree at all that education is key, although I personally believe we should focus far more on the trades than college (have you tried to hire an electrician in the last 10 years!). I just think there's a vast group of people who are essentially trapped in a position where they can't do anything but work two jobs to care for their families without assistance. The impact of putting our finger on the supply and demand curve to set a minimum wage has historically had little or even positive impact on unemployment while greatly helping these individuals, and the company's that employ them all seem to do just fine as well. It seems like decent policy while we work to enact more structural changes?
