My first job paid me $4 per hour, plus tips. That was 16 years ago and I was overpaid. However, that's what my labor was worth to my superiors, and they knew their business better than did the USG.
You make two assumptions with your comment. One is that everybody making minimum wage is supporting a family of four. I have sympathy for those people. Pricing them out of a job, however, is simply cruel. They need to increase the value of their skill with experience or education. And what better way to do that than to make less than they want for a period of time so that they can build up one or both of those.
Which brings me to your other assumption -- people who make a low wage will always make a low wage. I now make many times over that $4 per hour. That job enabled me to get another job which enabled me to get another job, etc. I wasn't doing what I do now, but every job had in common the attribute that I had to make myself worth what I was being paid, whether that's $4 per hour or $100. Every day I use experience gained at each of those jobs, even the most miserable ones.
There was a story on HN a while ago about a guy with an MBA who now works at Publix. He had been laid off and claimed that he wanted to teach his sons that work was dignified no matter the job. I believe he was being paid $8/hour or so. Publix has no choice but to pay him $8/hour because he likely can't do a much better job at bagging groceries than the 16 year-old next to him whose labor has been assigned that value by politicians who have never run a grocery store. If Publix wasn't forced to pay the teenager a higher wage than he's likely worth, Publix could either hire more teenagers and reduce everyone's workload and provide the customer a better experience, or they could pay the teen $4/hr and the father $12/hr.
That may enable the father to work there a little longer only to discover he is next in line for management because of his experience. His wife is thrilled that he's not traveling the world anymore eating out. And who better to run the store than someone who has been at the lowest level? Is the teen a slave at $4/hr? No, because he can walk next door to Kroger and see what they pay. Maybe it's $4.10. He decides to stay at Publix because he's already decorated his locker and made friends with the older guy who has great stories about traveling the world.
Businesses will determine quite well what their labor is worth and doesn't need USG to do it. Workers will determine quite well what corners they are willing to cut when times get hard. The latter may find that getting out of bed for a crappy, low-paying job enabled them to find a better opportunity.