It's possible that immigrants will vote to destroy the system that attracted them, but unlikely. Immigrants come here because they prefer life here to life at home. It wouldn't take a marketing genius to win them over to the cause of American liberty.
Commenter Pat nailed the problem with Bryan's statement:
So much for the Myth of the Rational Voter.
Just because some immigrant wants more freedom than they [sic] have in their awful country doesn't mean they [sic] won't vote to make our system less free.
When people come here from other countries, sometimes it's because they want more freedom and sometimes it's because they want more wealth. They see the possibilities for wealth but don't necessarily understand what political and economic system led to that wealth. When I moved to Winnipeg from a small town in Manitoba in 1967 and became a libertarian shortly after, I noticed this. I would meet people who came to Canada from England, which was a substantially less-economically-free country than Canada back then, and they voted for the New Democratic Party, the socialist party in Canada. They weren't dumb people. They just had no idea about the connection between the results in England that they disliked and the system that led to those results.
My reaction to Bryan's statement about how easy it would be to convince people is like George Stigler's reaction to a similar statement by Adam Smith. Stigler quoted Smith's famous passage:
I have never known much good done by those who affected to trade for the public good. It is an affectation, indeed, not very common among merchants, and very few words need be employed in dissuading them from it.
Stigler commented, "I wonder what those very few words were."
Similarly, I'd like to meet Bryan's marketing non-genius.