"Debates about minimum wages are most relevant to those closest to poverty. "

True.  A minimum wage REMOVES freedom from the laborer.  They are no long allowed to "bid" for work. If she or he wants the work they can be undercut by some pompous bureaucrat or clean finger nailed clergy member. 

Low level, first rung positions...where work virtues are learned....get cut.

As the (laid off) restaurant employees on the left coast are finding out....a 15 dollar minimum wage priced them out of work...just pushed them on to welfare, at taxpayer expense.   But the clergy can feel high and mighty.