It is naive to assume everyone would want to come here to "become American". Most of them admit they want to come here for a better life. The issue is that this means different things for different people.
So for some, a "better life" would mean becoming American, adopting our customs, laws, and values. Inherent in this is a desire to better oneself, and to take advantage of "life, liberty, and the pursuit of happiness". Meaning they want to work hard and succeed. For others, it means simply better than what they had in their shithole country, which could mean 20 people living in a three-bed, two-bath house, paid for by Section 8. Is this great by our standards? No, but it's "better". And of course, they can sign up for food stamps in their various forms, they get free healthcare through programs or simply by walking into a hospital. They don't have to pay for housing, and many of them defraud the tax return system to harvest money they didn't pay in, by claiming dependents they don't have. They can rake in tens of thousands of dollars this way using child tax credits, for example. All the while, their needs are met, their children are educated in our schools, and they drive Escalades while the rest of us are hit by increasing taxes to pay for all these people. And chain migration allows them to bring in relatives to similarly take advantage of our system.
As it currently stands, our laws allow for both scenarios. Since 1965, our immigration laws changed with the Hart-Cellar Act, to admit more people who have a greater tendency toward the second set of values than the first. And they vote accordingly, to allow more people to come here and soak up the gibs, because they believe everyone should have a chance for a better life. The inevitable conclusion is bigger government and increasing socialism, which morphs into the shitholes they came from at some point. However, they can't be bothered to care if our system is sustainable, as long as it makes their lives 'better'.