Slipshod, I think you say it yourself:
"So while proper integration is vital for a productive school, I am not sure what problem people have with overseas students."
Not all overseas parents, or their children, want integration. Some want results and a pathway through to tertiary education in an Anglophone country. When I worked in SE Asia many Chinese parents were worried about political instability and saw an overseas education an important investment not just as a passport to a better job but potentially a passport away from SE Asia. Many of these children would have been on "family scholarships". The extended family would find the money, then once the child graduated they too would be expected to help the next member of the family study abroad.
People seemed fairly divided on whether it was better to aim for a country like Canada which was seen as promoting multiculturalism and hence meant a child could retain their own culture, or somewhere like Australia which was perceived as promoting integration, which might mean better long term acceptance but some loss of cultural identity.
The pressure on the child to study hard and do well was huge. Some within this cohort, like in any other cohort, will be open and want to mix. Others may feel more comfortable mixing with others who understand both where they come from and the reasons behind their work ethic. This avoids conflict between family expectations, and expectations of peers and/or school in terms of spending time on non academic activities.
I would be very surprised if many in modern Britain have a general problem with overseas students. However if a child is going to board it is worth considering what they want from boarding and if others within that year group are likely to have the same expectations.
The group I have described are only one demographic. I hear that some schools are seeing a very different group of very rich young mainland Chinese who are sent over to the UK as a form of finishing school and to learn English, and who do not conform to any standard stereotype about Chinese being hard working.
My own children have benefited from having access to really international education on their doorstep, and have friends from all sorts of backgrounds. However there are potential downsides, and to work well schools may need to actively manage the process of ensuring integration. It is something a potential boarding parent should consider.