One problem with your analogy is that you're comparing legal immigrants coming here at a time we actually needed them.
And very importantly, Immigrants that wanted to be Americans. Compared with illegal immigrants coming here we don't need that have no desire to assimilate or even learn our language. For the most part they aren't paying income taxes but are using tax payer funded services.
Are you sure about being not needed? Are Americans clamoring for more $7/hr dishwasher/busboy/wait staff/fruit picking/landscaping jobs? Will that income be sufficient to enable them to buy houses, feed and clothe their families, save for college, and buy the goodies associated with "The American Way of Life (TM)" - eating out every night, new car(s), entertainment systems in the home and car, smart phones with 4G access, computers, 1000 channel cable TV, spa days for the wife and daughters, trips to the ball game, etc?
Nope, it isn't. The reality is, every culture has an underclass, be it slaves or immigrants, to provide the labor for the services citizens don't want to perform themselves. Citizen's have needs and wants requiring higher incomes.
You could argue that immigrants are taking jobs away from teenagers in the low-end service industries. Besides a large number being lazy, slow, unmotivated, and spoiled, they're also unavailable when you need them the most.
Yes, some immigrants do not pay taxes. Some non-payers use taxpayer-funded services. But in the aggregate, I bet they provide more in $ value to the American economy than they use.
Unlike "The Beast's" 3-day shopping trip to Italy and trips to Hawaii...
It's an intractable problem with no right answer. But my perspective is that the upside for the American economy and culture is far greater than the downside.
I know the right answer isn't locking down the borders so nobody can get in or out. Then our economy would experience 1 of 2 bad results - deflate as more people are forced to take lower-paying, less productive jobs. Or inflate as the workers price their labor according to their needs, causing everything up the chain to inflate to a greater degree to compensate.
Either way, things stay the same - upper class, middle class, lower class. Only the $$$ amounts differ.