When we moved to Luton in the 80s from the north the area I live in was mixed. White people, Asians and Caribbean people. Slowly but surely the white people have moved away. Many reasons for this and that's a big discussion in itself. But they have sold up and moved to outskirts of Luton or villages. My kids now have less chance of integrating and only go to school with 98% migrant kids. Fortunately I can afford to move to a more affluent area that has a more mixed area so my kids can grow up knowing people from all sorts of backgrounds. Not all can.
My main point is that integration is two way. You can't mix it with people when they don't want to. You have migrants who don't want to integrate and you get "indigenous" folk who don't want to either. Whose fault is it?
BOOM. The phenomenan of 'white flight'. You're totally right. This is why large-scale immigration does not work. This is a totally non-fact based estimate (BTW) but i'd honestly say if you took any sample of any group of humans, about 60-80% would not want to mix with anyone different. About another 30-15% would do so, but might find it difficult. The rest are those that actively embrace differences.
It even goes down to friends level. I have a small circle of friends, who share my interests, I feel comfortable with and get on with. The more different someone is to you the more awkward you generally feel around them.
All over the country there is proof that large-scale immigration does not work. Integration of a large number of people of a different background in the same area with a much larger number of people that already live there will take decades, or even maybe prove to be impossible.