Legal immigrants to America for over a century accepted and complied with these legal processes, many of which were even more burdensome or discriminatory (quotas, Chinese Exclusion Act, Immigration Act of 1924, etc).
It's unamerican to undermine a core tenant of the US's national sovereignty: the "sovereign right of States to determine their national migration policy", by arguing that unregulated, irregular migration is the norm and that any action to enforce immigration law is unthinkable.
For basically the entire first 80% of our history, the most the federal government would do in regards to immigration was write down someone's name and nationality, then send them on their way. You're right that in the early 1900s, there were finally some restrictions passed, but by that point the national fabric of America was already sewn, and even those restrictions only applied to specific countries. Right up to the modern era, millions and millions of people have been moving to the USA, and only very, very recently has there been an extremely formalized process, or efforts to go out of the way to deport people that aren't committing actual crimes (overstaying visas is a modern thing, and isn't the kind of crime I'm talking about) (and before you get on me about breaking the law making someone a criminal, lemme know how many times you've driven over the speed limit, and whether you verify your turn signals, headlights, and brakelights work every time you operate a motor vehicle).
I grant you that the government is often at odds at our values, such as with the Chinese exclusion act or when they put Japanese in internment camps, but the American culture has been pro-immigration and pro-refugee for our whole history. "Give me your tired, your poor, Your huddled masses..." No way in hell bureaucracy is more important to us than that value.
American conservative media are really good at repainting history, so they seem to have convinced some people such as yourself that American sovereignty is predicated upon a tightly controlled and monitored border, but that's basically the opposite of the truth of our history. My guess is that the conservatives need scapegoats to distract from the collapse of the living conditions of the working class from the true reasons (higher concentration of wealth) and so they've picked immigrants this time, a typical target. What genuinely surprises me is that people on hacker news, who I consider typically more media-savvy than the average person, are falling for this as well.
Anyway maybe take a quick scan of the wikipedia article to get a better understanding of our history in relation to immigration: https://en.wikipedia.org/wiki/History_of_immigration_to_the_...
If you want to go back to a pre-1900s American system and return to that national fabric, sure! Repeal the NFA and free our 2nd amendment. Repeal the income tax and welfare state, and go back to funding the government via tariffs and excise taxes (the liberal media sure convinced some people tariffs were an anathema to America, right?).
Your idealized “America the melting pot” thrived on the basis of people actually working for what they had, both immigrant and 3rd generation American; rather than sitting around drawing Social Security, disability, Medicare, Medicaid, SNAP, WIC,TANF, EITC, and housing assistance. I think you’re on to something!