My views on welfare can be summed up as "Government has the responsibility to provide what individual citizens cannot do for themselves".
Let's isolate this bit and pick it apart a bit.
So you believe government has the responsibility to provide what citizens cannot do for themselves. Presumably this means something like a minimum standard of living. So government, through tax revenues, has the responsibility to bring up citizens who fall below that minimum to that minimum (I assume you'd put a caveat on there that they'd have to be willing to work, not just lazy, etc. Not operative for this discussion).
Take, then, the example of a person working a minimum wage job at Walmart. You're saying it's the government's job to provide for that individual the difference between what he makes and what we've decided to define as the minimum standard of living. That difference is to be made up out of tax revenues, distributed over the whole tax base, presumably. But in the meantime, the beneficiary is the employer, who gets to pocket the difference between what they pay the employee and the minimum standard of living. The government basically agrees to subsidize wages so that low-wage employers don't have to worry at all about providing their workers a living wage.
You might argue that in a free market, those workers' wages have been determined by the market and who are we to impose controls on wages? But the implicit backstop of wage support for low wage workers acts as a distortion. Low wage employers know it's there.
