I am curious to know where you have lived in the USA.
I grew up on Long Island, NY and you could still sort of see the different pockets of immigrants that lived there when I was growing up. The Jewish tended to live in one part of town and the Italian Americans in another. The Chinese clustered in New York City so tightly that Chinatown became a thing.
They spoke their mother tongue and probably a lot less English than your parents did. But those were different times and the important question is not whether they, themselves, assimilated but rather… what was the next generation like?
I am a First Generation Asian and English has been my primary language since about age 4. My parents wanted me to learn my mother tongue but really I kind of rejected it. Which is too bad for me now but makes a point nicely…
We should not exactly care about desire to learn and speak English as an indicator of compatibility with this country. That will sort itself out in a generation or two. I, a sample set of one, am proof. :p. But really there are many like me.
As you say, we have a right to control the inflow of people, and we should definitely give ourselves permission to tread carefully where there is even a shred of concern. And fuck people who would call us racist for that.
But I find that I am less concerned with questions of assimilation than:
1. Whether a person is violently opposed to the principles of the United States and to rights as such
2. Whether a person is going to work or is merely coming to exploit welfare entitlements.
This is very hard to suss out in the short process of immigration. But easier to come to understand during the lengthy process of naturalization. And maybe the solution is to prioritize letting people in who wish to become Citizens and measuring these factors as a part of that process.