"The goal of a minimum wage is difficult to pin down"
In the most general sense it's straightforward.
Your salary is a function of your economic power vis-a-vis your employer - and nothing else. That power is usually derived as a function of the necessity (i.e. value creation) and scarcity of your skills, and possibly some other non-market forces.
Skilled workers have some modicum of leverage and can get a decent salary. Or have powerful guilds (like doctors).
'Capital class' will use their status as major asset holders, and non-market things like class, relationships, political patronage, lobbying, direct control of political systems etc. to manage their careers.
But unskilled workers have a problem. Basically, if there was no minimum wage (or other social intervention), workers would be paid the absolute minimum necessary to survive, and economies to support those very low levels would develop (i.e. 'Dollar Store' would be the '25 cent store) and there'd be favela like living conditions just as they already exist in advanced economies like Hong Kong and almost advanced economies like Brazil. (Obviously, if there is 'welfare' than the amount of welfare comp. would set the floor on this, i.e. 'work for the government and do nothing' would be the 'lowest alternative' to paid work.)
But especially without welfare, or non-market interventions in general ... that floor would be very, very low. Just take a trip outside of the 'rich world', it's easy to find workers working for 'a bowl of rice a day' whilst owners make considerably more, and by that I mean, there are definitely surpluses in the system to allow for higher salaries, but pure market power doesn't provide any impetus at all for employers to provide it.
So - 'minimum wage' is a non-market intervention, much like a 'union without membership' to establish a kind of meta collective bargaining.
Non market interventions are basically essential and they exist in one way or another in all advanced nations (either social services, welfare, min. wage. etc..). Should note that some argue that social intervention is correlated but not causal to wealth creation, but I disagree with this ... but that's for another discussion.