Elasticity in the US labor market is much closer to 0 than -1. Not only has this been studied, it's been studied enough that we can do pretty solid meta-studies to find bias is the studies:

http://cepr.net/blogs/cepr-blog/studying-the-studies-on-the-...

I wouldn't advocate a jump straight to a $15 minimum wage because for sufficiently large MW jumps the elasticity has to start heading for -1 (and beyond -- that's just the break even point) and we only have good information for small changes. But for small changes the evidence is unambiguous. The traditional narrative taught in Econ 101 is an oversimplification.

It's not hard to figure out why: markets set the actual price of labor between the producer price (minimum people will work for) and consumer price (value to the employer) based on bargaining power. If employers are winning at the bargaining power game, wages will be close to the minimum. If employees are winning, wages will be close to the maximum. Only those earning wages already close to the maximum will be affected by a minimum wage increase. Since the unskilled labor market is flooded, approximately everyone is on the "minimum" side of the spectrum. Both theory and evidence say that small MW increases are boons to laborers under these conditions.

As machines advance and begin push humans out of the market, the maximum side of the spectrum will begin to drift downwards and impinge on all the people crunched up against the minimum side, squeezing them out of the market and into "unnecessariat" status. Minimum wage will become an ineffective tool at that point, just as you say. But we aren't there yet.