I would say that, in general, it is not the academic attainment/ potential of other pupils in a school with a difficult catchment that affects the quality of education / experience provided.
It is all the things that cone along with the catchment. The high levels of poverty, poor housing, poor food, levels of addiction and crime and violence, family breakdown and dysfunction including abuse, poor employment options, high levels of SEN, low levels of parental education etc etc. In the increasing absence of other support serviced, school have to deal with all of these for their pupils.
It is not funding for schools directly that will improve this. Funding for healthcare, social services, policing, housing, family support workers, transport, to entice new employers into the area etc etc.
The main admissions change I would make is for every school in an area to be required to admit the average level of SEN and Pupil Premium children for that area, and for there to be a cap (like the class size cap) on the number of such pupils any school should have. So for example, if a district council has 4 pr 5 secondary schools and average PP of 25% and SEN of 15%, every school must admit 25% and 15% (supported by free school transport) and no school could be required to go above say 30% and 17%. This would both dilute the effect of some schools having to cope with disproportionate numbers and make sure that all schools have somewhat more equal cohorts.