Probably should look towards countries that do have a $15 an hour minimum wage like Australia.
1) No one leaves tips.
2) It enables people to pay for College/University to open up new possibilities for a career.
3) Higher wages, more disposable income, people spend more money, the wheels of the economy turn.
4) Other jobs that are seen as more "valuable" typically increase in the amount of wages paid.
5) You only pay an extra 50-70 cents for a Big Mac, whilst opening up possibilities for those workers.
6) American minimum wages have been stagnant for decades, despite inflation, a dollar today is worth less than a dollar 10 years ago... Thus an hours work today is worth less than it is 10 years ago. - In 1970 the minimum wage was $1.60 an hour. Today if you account for inflation... Would be worth $10.71. Instead it's $7-$9 in most states. Why? Because a few people are worried about a few cents extra on unhealthy food?
I've long believed that capitalism has some major flaws. For starters, if the person who owns the business can pay workers diddly to maximize profits, why wouldn't they? If they can continue to rake in more and more, and even raise prices along the way, but keep income low, more and more money spills into their pockets.
And to your points about more expensive Big Macs/etc, I simply cannot see how that'd be a problem for ANYONE. We went from $4/gal gas to $1.80/gal and also went the other way around. Several years back, it would cost close to $80 or more to fill a full sized truck, and people still did it. Now it costs much closer to $40. Those fluctuations in price may sway the opinion or choice of a minority of a group, but by and large people are stilling going to buy what they can when they can. And if they have more money to do it, more stuff will be bought.
My wife pointed out the other day that they did the stimulus package a while ago, and most Americans simply paid off debt so not a lot went into the economy. And the rationale being that giving the lower/middle class more money won't necessarily boost the economy. My thought there is simply that they were already getting screwed over for years and many started building a collection of debt to get by, and did the reasonably smart thing that they could by paying it off when they got that money. The country really needs years of decent wages for the lower and lower/middle class to see an economic boost. If you keep giving wealthy people more money... well, they'll spend some, but they'll likely save quite a bit also.
At the end of the day, a low minimum wage simply ensures that the upper percentile can continue to maximize their income on their business. To some extent, I see that as being necessary for them to keep the business moving forward. But to a larger extent, it really just seems like all anyone with money does is maximize their income while shitting on whoever they can to do it.
Hell, I'm not really in that lower/middle class, and I am seeing some of the same from the business I work for. Largely successful, highly profitable, but structured raises of 3%. I shouldn't bitch about 3% cause it is likely more than others get, but at the same time, you don't boast about record years and continue to hand out nothing in terms of raises and expect people to be happy about it.