The purpose of a minimum wage should not be, and is not, the minimum wage needed to live a comfortable and simple life on ones own, let alone supporting a family.
If one is unable or unwilling to get a position above minimum wage then they should be looking into ways to function based on that....search for another minimum wage individual also in a similar situation and rent a place together. Look towards family or friends if one has them. If not, look towards various non-profit groups and organizations where one can potentially meet other individuals. Change ones thinking to looking at perhaps renting a basement or a room from someone rather than having your own two bed room apartment. Etc.
The only areas of the country where it's likely neigh impossible for the average person to survive on ones own, on 40 total hours of minimum wage work, would likely be your metropolitan time places where the cost of living is drastically higher. Then again, in such a location, the amount of public transpertation, free forms of entertainment and engagement, ability to meet others, and social services provided are also significantly higher as well.
I also commented based off the notion of no further assistance. Which doesn't exist. We have rent subsidized housing, low income health care programs and subsidized free clinics, food stamp programs, various assistance subsidies with regards to transportation or communication, and it goes on and on.
I disagree that ALL jobs should pay, at minimum, the amount needed to live a comfortable life for oneself let alone a family. While I do respect the notion that there may be some rational tweaks to it that could be done, such as perhaps accounting for the higher cost of living in some areas, over all I think that the minimum wage is as it should be.