We're not governed by poetry on statues. If we decide we're full, that's ok.
There is a lot of what went into America that would be (or has been) lost by completely abandoning the idea though. Maybe its worth it, that's not for any one person to decide, but its a huge change and there's always the risk of throwing out the baby with the bath water.
Now with all the various welfare programmes you can easily become a cost, although of course that's only a proportion of the people coming in, just as it's only a proportion of the people born in the country. But it's silly to equate what happened back then with what's happening now. And it's unfair to hold the US to this standard and not any other country. Particularly when the US takes in more (legal) immigrants than any other country each year.
I don't hold other countries to a standard only because I have no say in how the are run. I would actually prefer to see countries freely allowing immigration and emigration, but I'm not going to say anyone else has to or even should.
What's often lost, in my opinion, with regards to the statue is that the whole point was that anyone should be able to move here if they want to give it a shot and none should be able to take away that right from anyone else.
Landmarks don't have to have legal force. My only point here, that I'm almost certainly over complicating or muddying, is that the statue is still held as a symbol of our country but we seem to fundamentally disagree with what the statue stood for in the first place.
It seems disingenuous but that alone isn't a huge deal. My concern personally is that I do still believe in what the statue stood for, and I think most people arguing immigration miss the fundamental question of should one person be allowed to stop another from choosing to live here and make a life for themselves. The argument usually jumps straight to how to limit it, not whether one should have the right to limit it.
Illegal immigration is a whole bag of worms since it only comes down to what laws we choose to write. Our immigration policies today are much more strict than they were historically (with a possible caveat for wartime years). Where it once was legal as long as you went through a port of entry and provided basic info, today its a multi year process tied to merit and limited by caps on total immigration and country of origin. Both were legal if you followed the rules of the day, but the rules are drastically more strict today.
We've got both a declining birth rate and an economy that demands growth. Our biggest competitor is 4x our size. We have vast open spaces, and a history of repeatedly, successfully incorporating waves of immigrants. We're not even remotely full.
Should we be able to decide who has a chance to live here at all? We have laws and you have to abide by them, but beyond that I'm not sure why I should get to be gate keeper at all. We'd have to get rid of welfare programs if we have no cap on immigrations, but honestly if someone wants to move here and try to make a better life for themselves I don't see what the problem is.
I'm sure most would say crime, but that's just not a compelling argument. We will never have a bulletproof immigration system, people will find ways to sneak through went we don't want to let them in. More importantly, crime isn't a foreign problem - we have domestic crime that has nothing to do with immigrants.
So let me rephrase: "we have already some internal crime so there's no point trying to limit another source of crime" ?
I've got some bad news for you, about babies.
When it comes to crime, its not that we already have crime so don't bother. We have to deal with crime either way, and the potential benefit of closing borders isn't worth stepping on what I consider a fundamental right. We also haven't answered how to know who will commit a crime if they come here, unless we have precogs hidden away in a pool somewhere.