There's two parts to that.
First, there 100% were segregated, publically un assimilated migrant areas, they are just far enough back that no one currently living remembers them.
Five Points, the various Chinatowns, the German neighborhoods in the midwest, all of them would frequently have languages other than English as the main language well into the 20th century. The Aisan enclaves still do. I can easily find markets and neighborhoods where signs are Korean, and if I want to buy something we're doing it through a grandkid or Google translate.
I think it's easy to look at the results of the Third wave of immigration (1880-1914) and see successful assimilation, and overlook we are almost 150 years on from the beginning. It's also true that assimilation was not 1 way in any of the immigration waves, so concern that "base" American culture (whatever that is) will change is probably accurate. "Base" culture has shifted on each of the four major waves of immigration. How exactly that shakes out is harder to predict.
On your second point of them forcing their culture on others, or trying to stop us from doing something we want, that's an interesting take. If forcing their culture on us is openly practicing their religion, previous immigrants undoubtedly did that. To great cultural friction, most notably the Catholics of the second wave. If it's opening businesses and selling *expletive deleted*it from their culture I don't know what to tell you.
If you are referring to the push to downplay Thanksgiving and Christian holidays and include every random holiday on the calendar in schools and shops.....that's not immigrants pushing that. That's the AWFLs. I might (and do) see a muslim family hang a crescent in their window for Eid, but they aren't down at the school board meeting demanding Ramadan celebrations in schools. That's the White folks overeager to prove they aren't racist.
Same with Ireland in this thread. The folks strengthening hate speech laws and forcing the Irish to sit there and take it are other white Irish in a tizzy to signal to other white Europeans their virtues.