You are wrong. Data shows that hard working unskilled immigrants are net payers into the social system. And they are a big positive to the economy overall. America became the world's superpower on the back of unskilled immigrants.
Today's economy and the economy in 1920 or even 1960 don't much resemble each other. For starters, the costs of labor regulation and employer mandates sets a pretty high price floor on labor. That creates a much more difficult market for workers with relatively low marginal productivity to find work.
Also, the population has a far more top-heavy age distribution than it used to have. This country is going to need a very productive labor force to maintain the welfare state at anything close to the current tax levels.
[1] http://c4.nrostatic.com/sites/default/files/YL%20chart.jpg