Is it the basis of our society to be accepting of all cultures? Since when?
How many places with a strong cultural identity have a liberal approach to immigration? Do the two ever go together? I've asked before for examples of such places. I don't think that they exist.
What is a strong cultural identity? If you mean somewhere like Japan, then that is because they have Jus sanguinis citizenship policy.
I'd argue that the USA, Canada and France all have quite a liberal approach to immigration. France certainly has a strong cultural identity, that is if you count eating well and drinking wine as culture.
If you believe we had a strong cultural identity before the waves of immigration since the 50's please define it.