Your question is very interesting.
My late father came here illegally around 1949. He and his brother crossed the Canadian/American border with his new car he purchased after working in Canada for a few year (waiting there to get papers in order to immigrate to the US.)
My father got a high paying job in the defense industry in San Fran. One day he came home and INS officers arrested him. His brother had escaped with the car and drove back to Canada.
My father spent a day in custody. The next day the INS officers gave him an ultimatum. He could voluntarily go back and get back in line or he can refuse to go back and get deported. He chose the former. Seven years later he entered the US legally.
My father and mother each had $3000 in their bank accounts when they came over. Those were the days when we had a competent government. FYI, the border guards had taken down the license plate number when they crossed the border. When they didn't return the INS searched for their car and arrested my father as he was coming home from work.
The idea that people in their natural state are a burden is a leftist, collectivist mentality.
Free people are an asset to a society. I believe that all men have the capacity to be free though too often they are convinced to give up their freedom for the "security" of slavery.
If someone has a problem with immigrants coming to this country and abusing the welfare state, then fix the welfare state. Don't keep out productive human beings who want to better their lives because people engaging in such behavior will better everyone's lives.