This assumes that everyone is in a similar situation in terms of trying to support a particular kind of lifestyle on whatever is deemed to be the minimum wage (or maybe two of them).
Should those same considerations apply to teenagers who are not trying to support that lifestyle themselves? What about the ones who _are_, including emancipated minors?
What about retirees who are working part time because they like to feel useful but generally not expecting to live off that money?
It feels to me that the right solution to the problem of us feeling that people in a certain situation (e.g. supporting kids) should have a certain minimum income is probably better solved via solutions like the EITC (which can be much more tailored based on overall financial situation), not a minimum wage.
Of course that gets some people arguing about the dignity of work and how receiving handouts rots people's souls. But I would argue that taxes are opaque enough that people receiving EITC don't perceive it as a "handout" anyway....