There's nothing arbitrary about paying a living wage. I could rephrase your argument as "Shouldn't businesses be able to pay poverty wages, and taxpayers can make up the difference?" If you can't stay in business without the government supporting your workers, then your business doesn't work. These businesses are effectively being subsidized by the government without giving the government a stake.

Also, I feel like this isn't even worth addressing, but no, work is not a government-sponsored day care for workers.