That only works if you have a very, very long time horizon, something that teens are known for, right? Once you add up foregone earnings and the cost of college tuition (even at a state college), you're down about $160k even before you graduate college. After that, you're probably not earning much more in your first few years than a person who has 4 years of work experience. At a conservative estimate, you're going to start making a dent on that $160k about 6 years after graduating high school (incidentally, why graduate high school?). Let's say that at this point, you make $10k more per year, a number that increases by a thousand or two each year. It would take another decade before you catch up. So either you invest 4 years into a college degree and not see a real return for 16 years or you simply try to make a career flipping burgers. Sure, for the motivated amongst us or those with educated parents, that's not going to be a real choice. But for everyone else, particularly the poor? All this will do is worsen existing education disparities, which will cause even more inequality down the line.
In fact, the main reason for getting a college degree once this minimum wage hike comes into effect is to increase one's chances of getting a job, not to increase one's salary. Given that employers will make the same profit from workers who earn above $15 as they do now, the employment situation for those people will remain stable (in fact, it might improve a little bit to offset sharp decreases of employment for the unskilled). But who in their right mind will continue to hire people at $15 an hour when those workers only justify a wage of $9? Does anyone seriously believe that your average minimum wage business (i.e. retail store or fast food joint) actually make a 66% profit on the wages of every employee (after accounting for profits that are necessary to offset the risk of owning a store)? Who do you think will be the first to get fired? It's going to be ethnic minorities and the disabled.
Furthermore, on what basis are you going to hire $15 workers when the supply of those workers ends up far higher than demand? We're going to start seeing bribes just to get jobs, just like the shitty parts of the world. Can't wait to see that.