I don't buy this characterization, this misconstrues the entire foundation of the American experiment. The founders were not driven by some impetus to preserve European culture. There was nothing more pinnacle "European culture" than the monarchy itself and how it organized the entirety of society.

Among the founders themselves there was Alexander Hamilton. Zero familial or cultural connections to the group of EUROPEAN settlers that formed the United States. Zero social standing in late 18th century aristocratic social classes of the colonies. A West Indies immigrant (being an illegitimate child with murkiness about parentage, he may not have been 100% Caucasian), but quickly rose in the ranks by sheer effort and ambition. He was eventually harangued throughout his short life (mainly by fake-news Jeffersonians) as somehow not being genuinely loyal to the republic because of his foreign, lower social class background, in addition to his federalist political philosophy.

Still what Hamilton proved from the beginning was that bloodline, social and cultural standing didn't matter if government could adopt principles of a meritocracy. Educated leaders no matter their background or social class.

so they inevitably ends up voting in massive numbers for the politicians that promise them the most free stuff and the best opportunities for them to bring the rest of their relatives into the country, where they too can vote for more free stuff and more relatives to come in, ad nauseum.

This is in no way unique to legal immigrants, this is how any politician tries to pander to the native public, their constituents. It's whenever those in power focus on what will please, rather than what will benefit the people. Legitimizing and embellishing perceived grievances + promising a free ticket in some way. That's why pure democracy was anathema to the founders, it just leads to catering to popular will even at the expense of the public good.