While I agree with the mentality that no one is entitled to a wage (and that a person is worth more as an individual than their paycheck), I feel it's not unreasonable to ensure that people who do work, and work hard, aren't being shortchanged by their employers. While it is largely possible to negotiate pay raises (people these days are generally pretty decent) or switch jobs, there absolutely are situations where a worker may be taken advantage of.
One idea I've been bouncing around in my head, though I've no idea how it'd work, is to not only have the aforementioned dependency-status impact on pay, but let the employer themselves decided whether a position should be considered one 'career-worthy' or not. If not, then that position would not be subject to minimum wage requirements and it's purely a position in which the relationship between employee and employer (as you described) reigns supreme. If they do want to meet minimum wage requirements though then they should qualify for a tax credit of equal value as the increase cost in payroll. Ideally that incentivises businesses to increase pay for employees while keeping their costs from unnecessarily ballooning. Is it practical or necessarily the right solution? I dunno...I'm not a policymaker and I've not really thought this through. However I think it's a far better alternative than either relaxing the minimum wage requirement entirely or raising it to an absurd level like the oft-clamored for $15 an hour.