RockaLock
It boosts the school because it means the overall attainment of their cohort increases, and it also means that the 'better' state schools will have an increasing number of families who value education highly and have the money to pay inflated house prices.
If I think about two different Year 7 classes I have had in different schools.
School A: Students come from high prior attaining primary schools. Tutors for SATS is quite normal. The feeder primaries have much fewer social issues to contend with that others in the same local authority. Students who join from prep school are often higher prior attaining too. This Year 7 class was almost entirely expected standard with a large proportion of greater depth. Some children are working towards, but they generally have SEN needs and because they're such a small number it's very easy as a classroom teacher to meet their needs. In a mixed ability class I am able to meet the needs to all students well in my subject. The children in this class have good oracy, arrive in Year 7 with strong reading skills, they're good at working with other children, they've got a range of soft skills that means they handle challenges well, and generally can resolve disputes well. The curriculum is challenging, moves quickly and in turn these children are likely to cover a lot between Year 7-11.
School B: Year 7 students come in with highly varied attainment levels. Most are below average, with many leaving primary school with a reading age of upper KS1/LKS2. These children have to access a secondary curriculum from this point. There's a high level of diagnosed and undiagnosed SEN, as well as a high prevalence of poor SEMH behaviours. The school employs parent and family workers as well as a huge pastoral team to pick up the never ending series of issues from outside of school (many of these issues are caused by problems accessing other services). In a class of 30 Year 7 students, only a tiny number come from households where reading is valued. Attendance in this school has been low for decades, just like previous schools in the catchment. Lots of time at the start of the year has to be spent teaching 11/12 year olds how to disagree without being rude, threatening, and verbally abusive. Many of the parents have a low view of schooling and a sizable minority are highly confrontational. On the whole the school is not only having to try to catch up children who are behind educationally, but they're having to try to break complex intergenerational cycles.
School B has a mixed history with it's standards, but locally it's accepted it's doing the best it can with the cohort and they're making some progress. Outcomes are low though.
In the event private schools close, those parents are going to move into School A's catchment. They're simply not going to send their children to School B. Why would they because the overall level of education is lower?
School A will be even more oversubscribed, School A families who lose out due to School A catchment prices astronomically rising will find their children sent elsewhere, (School C, D etc as parents select the next best schools). Nobody concerned about outcomes will choose School B. Some children whose parents haven't the means will be forced into School B, and they'll get a substandard education. Those with the means to home educate and pay for tutors will do that rather than send their child to School B.
Closing private schools isn't going to suddenly make School B better.