Earlier quoted context omitted.
Critical omission on your part: not everyone needs to live on this wage. Think about high school kids trying to get a summer job. They don't need to survive on this wage, they simply need something productive to do over the summer. Should my business be required to shell out $15/hr for a high school kid, when the value of his/her labor is way less than that?
How about homeless high school kids? Or high school kids with one parent who are helping to supplement their household income?
A high school kid making money regardless of his family's wealth is making money as a supplement to what the parents or guardians should already be receiving from the government or otherwise, assuming they aren't emancipated but that is a very special case. If the parents can't afford to meet the basic needs of their child despite how ever much welfare they receive then why should the state allow for a child to remain in that home?