I understand the sentiment, but the problem is that a rising minimum wage is guaranteed to result in rising prices for staple products, while the majority of the people purchasing them aren't earning any more.
Facile example: as a middle class guy, I can pay $5 for a box of blueberries and just have them in the house. $10 pushes it to be a treat that I would buy ten times less often. $20 makes it a luxury I buy once a year at most.
If we increase the minimum wage, and fight for people working illegally under the table for less than the minimum (which has to be part of the plan if you intend to have any honesty in your approach), the result is necessarily that the price of blueberries will rise. If this pushes the cost of blueberries to the point where normal people won't buy them anymore, the business model becomes unsustainable.
As the wage rises, during the process, we'll see worker conditions deteriorate, an increased turn to illegal labour (and a commensurate demand for illegal immigrants to do it, which furthers that problem), and eventually either farms folding, changing crops to something more profitable, or perhaps subsidies coming in. Of course, we should have learned from corn what happens when we oversubsidize crops; at the very least price discovery is somewhat removed from the market and so things are priced cheaply that shouldn't be, people overconsume while tax dollars make up the difference, etc. One can probably argue that staple crops (wheat, etc) should be subsidized, but blueberries aren't quite as critical, are they?
To make an example more germane to this particular issue: if the minimum wage increases and all of the people involved in the chain leading to these food delivery drivers costs too much, when the potential consumer sees that ordering food is simply too expensive for them, they'll just give up. Entire industries that function because intelligent people have carefully budgeted around wages and expenses to eke out a living will be negatively impacted, and parts of them will fail.
And yet, we'll survive it. But let's not kid ourselves: the standard of living for the poor isn't going to get any better by changing this number. Instead, what happens is that the middle class effectively earns less (try calculating your salary as a function of the minimum wage - if you made 3x before, maybe you make 2.5x now; if you made 1.25x before, maybe you make 1x now....) while the rich don't even notice the change. For brief periods of time while the market settles, those at the minimum wage line will have an advantage, but once the price of goods and services finishes rising to compensate, it's back to square one. The purchasing power of the dollar simply drops to match.