With a few exceptions, that's essentially how the current system works anyway. As I understand it, although there are no 'catchment areas', priority is given to whoever is closest to the school by drawing a straight line on a map, which in my mid is a 'catchment area'.
I'm not sure what the distance rules are (does anyone know?) but York does have catchment areas; the maps are available online.
Where I live, some of the parents are desperate for their kids to go to one of the religious schools as they are perceived as better. However in reality one of those schools actually has a big bullying problem according to various people I've spoken to (including parents and children), but of course it gets good results and has a rather impressive building, so everyone thinks it's wonderful.
Then there are two regular schools, but one of them supposedly has a better reputation than the other, but the reality (from speaking to staff who have experience of both) is that they are a lot more similar than some people believe. However many parents naturally want their children to attend the school that has more affluent areas in its catchment area, than the school that has more deprived areas in its catchment area. In practise though, having spoken to a dozen kids earlier this year, who did
not get their [parents] first choice of school, when I asked them the question, they were all really happy with the school they ended up with and felt they were no worse off than if they had got their first choice. Some of them even thought the opportunities they were given meant that they felt they had possibly a 'better' experience than their friends who went to the other school.
So a lot of the extra travelling to go to supposedly 'better' schools is driven by parents, rather than the children themselves!
...Forcing kids to go to their nearest school makes things worse not better. For schools close to run down estates, council estates, etc., it limits their choice and social mobility. A school with a catchment of "problem" homes is simply never going to succeed. You'd need to rebuild schools in specially selected areas with a decent mix of households for it to work - that's never going to happen.
I think in York there probably aren't any schools that don't have a mix in their catchment areas, so maybe the problem you talk of doesn't exist in York, but some do have more deprived areas than others. You can work it out by comparing
Datashine with the catchment area maps on the Council's website.
However you are right that schools with a substantial proportion of deprived areas in its catchment area can almost never be rated as outstanding by Ofsted and it's a struggle for them to be rated Good, but what you will find is that students who do want to succeed who attend such schools generally do well and are not held back, and that they go on to become well rounded, down to earth people.
You'll also find some thoroughly nice kids from deprived areas and there's not many things in life that are more rewarding than seeing them do well (not that there is any opportunity to help them at the moment - but that's another story, and off topic for this thread, so I won't go there!

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