That's a very shallow view of the market, pretending like moving one thing doesn't move everything else. If the price of labor goes down, the price of goods and services will shortly follow suit. That's why shit produced in China has been so cheap, labor is cheap.
>Perhaps reducing or eliminating the minimum wage would make employers more productive and more profitable;
It would increase the economic efficiency of the entire system. The employers are already productive and profitable, they just ship your job to China where labor is cheaper. Not only are you unemployed, but every moment you lack a job your market value is decreasing. Nobody wants to hire someone who has been unemployed for a long time. You're also not building useful skills, or a reputation/references for being a good worker.
Perhaps reducing or eliminating the minimum wage would make employers more productive and more profitable; they could then put that money back onto circulation, making life better for the entire economy. Or it could reinforce the concentration of wealth and put companies into the awkward position of not having anyone to sell their products to.