Sort of. You're kinda getting the puzzle together.
Welfare is designed to be a safety net for people so we don't have people starving in the streets. It's not meant to act as a wage subsidy so corporations don't have to pay their employees.
The minimum wage is supposed to be high enough that a person doesn't need food stamps if they work full time.
You should be able to see the problem we have today based on those two statements.
Things that help keep a decent share of new wealth going to the working folks and not just the already rich is strong labor laws, unions, and reinvestment of tax dollars into things that help people the most. Part of that money comes from the rich, but one issue we are having is a continuous stream of tax breaks going to the rich, a slow and steady assault on labor protections, securities, and workers unions.
I agree, the system right now is a bit skewed. It shouldn't take that much to fix it. But what we really need to avoid are tax loopholes. Trump didn't pay his taxes in a perfectly legal way, because of tax loopholes. Apple and Nike have millions due in taxes, but they keep their money overseas in 'tax havens' so that they don't have to pay. Just a few additions on to laws here and there, and then it will be illegal to try and use tax loopholes.