What they were buying into was the ethos of the US, democracy, toleration and above all the freedom to aspire to have a better life.  Nobody said they had to give up their religion or culture to become Americans.

Some of my American friends live in Pennsylvania, in a small town with a large Polish community that has a vibrant social life.  Those who want to are free to learn the Polish language, traditions, arts, crafts and cuisine.  No matter how much they keep their Polishness they come together on the 4th of July to celebrate America and being American.  They are very definitely Americans first and are fully integrated into its ethos.

The same with the Chinese, Irish, Italians, Jews and many others.  They all went there to get away from societies where they had no opportunity, were repressed or persecuted or just wanted a better life.