Instead of scrapping the system you could have people be qualified for it only if they either grew up there, or worked there for a period of time before hand. It is a good system.
This is the centrist position and I agree with it. Coincidentally, it's also Trump's position. More/better jobs solves the welfare problem. Immigrants can come if they want to work.
Which most do. People even joke about immigrants owning corner stores, but really it just shows the willingness to open a business. Willingness to take risks is a good thing.
