I'm skeptical about the claims that are being made about the size of payouts. They pertain to legal action and finding of liability. That's an important thing to take into consideration.
Also, there are better ways to welcome new Americans. Most of them don't need or really even want a handout, they want to work.
The primary problem is that existing migration quotas are artificially low, and so low that the harm extends to current American citizens.
An additional 40,000 truckers would be very helpful right about now.
The United States has sufficient resources and land to accommodate a billion lives, at least. That we currently limit migration is to our own detriment.
Americans ought to consider how their lives will be affected on net over the long term when thinking about migration policy.
Most people look at short term consequences to shifts in labor markets. Right now, of course, we have a bit of a labor shortage. But that could be related to a lot of things.
In the long run, new Americans are more likely than others to be entrepreneurs. New Americans create more new jobs than existing Americans, on average.
New Americans, if allowed to migrate legally (have to relax the quotas!) pay taxes and are on net contributors into social welfare programs rather than recipients. I'd be agreeable to allowing legal migration that excludes new Americans from social welfare programs if it would satisfy some people. But I don't think it should really make a difference.
I tend to discount concerns about cultural shifts that I understand you think are very important. Maybe that is a flaw in my reasoning. Perhaps because it is hard to quantify economists are prone to discounting it as well. On the other hand, many of the elements of culture that you might want to preserve are elements that I do not prefer. The latter has to do with internal disagreements among Americans.