The fallacy is that all people are equal and have equal opportunities. Pull yourself up by your bootstraps is what many Libertarians and Conservatives say. I'm not advocating that everyone be treated equally either. Someone who strives to do better whether it be by ability or drive should receive more. That does not mean that those with a lesser IQ or skill should be treated like dirt or not be given the opportunity to work for a living wage. People don't stay on welfare because they like it, they stay on welfare because it's not possible to live on $8/hr. Everyone can't own a business, or be an engineer or doctor. To believe that everyone in the US has the same opportunity when they are born is looking at the world through rose colored glasses.
What happens when you take away the ability to get a good meal or an education from those with less? They tend to have children who grow up in the same situation and the problem perpetuates.
Because very few people abuse the system let's just close the system down. That's just ignorance. Under that thinking, nothing would be allowed at any level whether it be private or public because there is some fraud and abuse in every sector of society.