The trouble is, the American values you describe aren't even shared universally across Americans, right here and now. There's a huge battle going on right now against the separation of religion and state. People are making laws based on Christian scripture. There are people who think prayer (but only Christian prayer) belongs in schools. And rationality? Were you here for the past four years? COVID brought out the biggest wave of irrational science deniers and people fighting against evidence-based decision making. And strangely, whether you're pro-vaccine or pro-ivermectin you can look at the other side and claim that they're the irrational ones refusing to believe the evidence.
But let's talk specifically about freedom. Sure, you won't find any American who says they're against freedom. But what does freedom mean, specifically? Very few people actually believe in absolute freedom - the freedom to do whatever I want, whenever I want, and to whomever I want. Most people would say I shouldn't be free to punch someone I disagree with. There's a huge rift right now over whether women should have the freedom of bodily autonomy regarding pregnancy, with a lot of the loudest advocates for freedom coming out against it. Or the freedom to choose your own pronouns and gender. The freedom to marry someone of the same sex. The freedom of people to smoke in public. In fact, your original post seemed to be opposed to the freedom to immigrate to this country. (At least, you stated that "I think collapse due to mass economic migration is more likely." That seems to be an argument against immigration, but who knows? Maybe you're pro-societal-collapse.)
Freedom is the quintessential American value - so vague that everyone can claim to be in favor of it while fighting tooth and nail to deny it to the people they don't agree with.
My point to all this is simply that "they don't share our values" is a not a valid reason to oppose immigration. There's no such thing as "our values". Half the people in this country already don't share the values of the other half. Or like "freedom", only share the ideal of it but not the practice. Immigration certainly isn't going to inject more chaos into what we already have.