If business can lower wages, people can work for less and get invaluable on the job training and having government control wages causes businesses to lose money affecting employees and stockholders so everyone now suffers. So it's better to let companies handle it because they know what's better for themselves.
A private sector with no regulation would be so disorganized it would be disastrous. Businesses need government regulation in order to operate ethically and with order. Wages need to be regulated because a business's primary purpose is to make money and cut costs when reasonably possible. A business's payroll account is often the costliest expenditure in a business's income statement. If their was no minimum wage set by a legislature that would encourage many companies to cut that expenditure to increase cash flow. That is the logical outcome if you eliminated minimum wage. Poverty would skyrocket growing the achievement gap even more. If anything you should increase the minimum wage because that would be even more beneficial in the long run for the society that the companies are in.
If there was no minimum wage and companies were free to pay workers as much or as little as they like then that would be extremely unfair. Businesses would all lower their wages making it extremely hard for workers to earn a living. The average wage would plummet without regulations as businesses would see no need to pay such high wages and their would be a universal lowering workers earnings, particularly at the lower levels. With the cost of living getting higher these days and countless families barely scrapping by, any regulation like this being put in place would cause mass unemployment and the standard of living to dramatically drop.