Two points:
I believe you are not entirely correct about “Any dilution of the existing population.”
I don’t think it is cultural, although all humans are to a degree products of culture.
I believe human beings are “predisposed to liberty” and that America has, in the past, attracted those. Our history is full of their stories. We still attract them, from Mexico and elsewhere.
However, America has changed to attracting too many predisposed to government dependence.
And, as I’ve said several times, even without immigrants we’re going down the same road with non-immigrants. In this regard illegal immigrants are a small fraction of the problem.
The root problem is we have created a government with the size and capability to create dependency.
The fly in the ointment is that immigrants vote to keep the welfare state, so the more you let in the harder it is to implement Friedman's solution.
Stopping immigration is a prerequisite of shrinking the welfare state. There is no getting around the immigration issue. If it isn't controlled the United States will end up at the global mean for liberty, a considerable decline.