But employers don't set the wage. If they did, then all wages would fall to the minimum wage, when in fact only 2% of the US labor force does. Why is that?
Your insight is correct, but I don't think you're following it through to its conclusion.
As a thought-experiment, step through the economic process that would occur in the face of increasing the minimum wage to $100. Think about how the prices would flow through the economy.
Nominal wages across the spectrum would increase at various rates, leading to a general price inflation. Actual wage increases, if any, would last only until the market adjusted to the new price levels.
In the meantime, the structure of the market would shift to become more capital-intensive and less labor-intensive, and would increase imports from labor-rich but capital-poor nations. Prices are information, and the market obeys. When you tell the market that labor is more expensive, it will adapt to need less labor or look for it elsewhere. It's not clear to me how such a scheme helps low-skilled workers.
Now, of course no one is arguing for a $100 minimum wage, but the economic effects of a smaller minimum wage are different only in magnitude.
The core flaw is not understanding that prices emerge from the market. When you try to treat them as inputs rather than outputs, you create systemic misinformation. That distortion tends to lead to misallocated resources which in turn leads to a poorer world.
Russ Roberts covers the answer better than I could: http://www.econlib.org/library/Columns/y2005/Robertsmarkets....