There are 3 comprehensives near me. One is very high achieving, very strict, has great value added scores and children on FSM and with SEND, on average, make as much progress as those from wealthy backgrounds. Another does OK academically. It is much less strict and sells itself as being more flexible and accommodating of difference. Classrooms are quite chaotic at times and there is quite a bit of disruption. The third school sits in between the other two in terms of culture/approach.
Children at the local primary tend to go to one of the three (with a handful going private). Different families are drawn to (select) the different schools for different reasons. There are lots of families that don't get their first choice and end up very disappointed by being allocated one of the others.
There are lots of misconceptions and false rumours about the high achieving school. It is strict but warm, it has high expectations but provides lots of support so expectations can be met. It has homework clubs where teachers and TAs provide support and resources so there is never an excuse for not completing homework. There are rails of donated uniforms so no child need not comply with uniform expectations. For those parents who work with the school, if/when their child starts going off track, together they sort it out. For those parents who aren't willing or able to work in partnership with the school, the children do end up with lots of detentions etc. The school make it very clear that they expect parents to be actively involved in addressing any issues that arise. This is what puts lots of parents off.
I recognise that parents who have had difficult school experiences in their own childhood find it difficult to trust schools or face being in them. I also recognise that the 'middle class' culture of schools feels alien/difficult for some. I also realise that their kids are likely to struggle the most in school. I can, therefore, see the attraction of a school that seems less intimidating. The thing is, the children don't do better in the 'less intimidating' school and those parents who can face the discomfort and do work with the high achieving school end up delighted with it - especially when making comparisons between their peers who went to the other schools.
If schools are well funded and run, the variable that can undermine this is pupil behaviour that is not linked to the education offer. If parents are unwilling to work with the school, these won't get addressed. OP's idea of diluting the impact by spreading them across schools would prevent some schools being overwhelmed by these issues and therefore unable to be able to offer effective education. However, it would also increase the burden elsewhere - not address the issue at hand.
The challenge is the expectation (current need) for schools to provide more than education. Many issue impacting on pupils are more 'social care' related than education related. Social care should be properly resourced to address the issues arising. Schools should have their own team of social workers to address issues arising. I don't think it is just funding that gets in the way. People don't want their parenting to be judged or challenged (there are lots of reasons for this). Instead they want schools to sort everything out, so they don't have to.