If a business is operating profitably by paying people wages below the cost of living, it is being subsidized by the government and therefore is not independently profitable in any meaningful sense.
I view the government as setting up a playing field which entrepreneurs than explore to find profit opportunities. With welfare and especially the Earned Income Tax Credit, the government created a class of people who can be profitably employed by paying $X below the cost of living. This was intentional, a way to keep people in the market economy even though they're not productive enough to survive in current conditions.
An entrepreneur who employs someone like this is like someone living in New York City in a rent-controlled apartment. You can't say they're not supporting themselves just because they couldn't afford rent in a pure market system. The actual system was purposely designed to make certain activities profitable that would ordinarily be losses. Without people taking advantage of those niches, the subsidies would be pointless.
The subsidies in this case do not exist so that businesses can exploit underpaid workers, though (and are not comparable to something like a rent-controlled apartment, where landlords are just prevented from charging so much that they'll drive out the workforce rather than actively subsidizing with things like food stamps or welfare). They exist because the US is reluctant to let people starve in the streets and die, the fact that you interpret them as an effort to "keep [unproductive] people in the market economy" notwithstanding.
The government bumping up unemployment to the point that people don't have to take certain types of jobs that rely on underpayment for profitability does not contradict that goal, which is why many people are not that sympathetic to small business owners complaining about it.
Employment is an explicit goal of the US welfare system. It's not supposed to keep you from starving if you don't work. That's why you are limited to three months of food stamps every three years unless you are working.
I do see rent control as directly comparable. Government observes a matrix of incomes and prices determined by self-interest in a market economy. Government determines some are not politically acceptable and modifies them. Now the military pays better than before, NYC rent is cheaper, and low-wage workers are more able to survive. Now people build businesses around contracting for the military, living in NYC, and hiring low-wage workers. If government later changes the payoffs to these activities, it's not the businesses' fault for relying on the prices the government set. Government set those prices to enable those businesses.
