I actually think that's a weak example. It reminds me of the appeals to emotion I saw about how "Our boys in uniform are fighting for our freedom for less than $15/hr and these fucking commies think that's what we should pay fry cooks!". Rather than denigrating fry cooks, maybe we should pay soldiers more. Similarly, rather than trying to keep the guy below me from making more, why wouldn't I push to make more for myself?
See, in my trade, as a crew lead I expect to make about double what I would pay a hard worker with zero relevant xp, and about 60-70% more than that same person after 1-3 years of experience. So, if you expect me to lead a crew of guys who at MINIMUM are going to be making $15/hr, you better be paying me >= $30/hr or I'll find someone that will. If I show up tomorrow and my current salary is the new starting wage, I'm not going to say, "Oh well, I guess I'll just abandon all sense of ambition and work ethic because some swing dick fresh off the street suddenly makes the same as me now.". I'm going to tell the boss to open his wallet, or I'm out the door, and he can let the lvl 1 swing dick run the crew. (BTW, this will result in failure.)
That said, I'm not arguing for a $15/hr minimum wage, I just don't think that was a good example of why it's a bad idea. The last place I worked, scaling to a $15/hr minimum would've bumped up our companies payroll by around 25%. That's not nothing. Absorbable in my opinion, as long as our competition has to do it too, but still. That's not acccounting for all our suppliers upping THEIR rates, thus increasing our costs in ways I don't have the math chops to accurately predict. Not to mention the effects of raising the minimum wage on inflation and COL as a whole, which I've read conflicting things about and can't form an educated opinion (I'm a level 10 swing dick, not an economist).
Lower margin industries could have it worse, and I could see some places closing before things equalized. I'm sure places like Wal-Mart could cover their increase in payroll without noticing it, but most places would feel it. So yeah, it might be a good idea is it might not be, but I think it's way more complicated than the, "Pay people more, so they'll spend more, so you make more." spiel you hear from the Bernie fans.
I do want to add, I recently moved to Seattle, where the minimum compensation is $15.45. I'm making what I expect to make, no-one at my company is butthurt over their pay and everything seems to be running along smoothly. I haven't really talked to the owners about how difficult it was to transition to that rate, but it's a small shop so from what I understand they would've done it incrementally. From an outsiders POV, the economic problems here seem to be based more on the lack of housing and out of control taxation (force companies to pay more, so we can tax more?) than the effects of a +$15/hr minimum wage. Excluding rent (which is a HUGE exclusion), COL here doesn't seem outrageous, and in some ways is cheaper than I'm used to. Dining out in my neighborhood is pricey though, which I imagine is mostly due to the rent/property taxes, but the min wave can't help. Then again, what do I know? I'm just an un-frozen caveman swing dick, and micro-economics confuses and scares me.