Moving for a 'good comp' isn't an education system, though, it is an admission system based on distance. Moving for primary school is very common but I see no one suggesting KS2 Grammars to solve that unfairness.
Some Grammars have distance criteria, some comps admit by lottery.
Maybe, just maybe, one of the factors that contributes to success (higher stats) in London schools is that the socio-economic sorting hat is not so emphatic. In London Georgian conservation areas back on to high density housing estates, I can't think of any schools (popular, high performing schools) near me which have a home-owner catchment that do not also have large LA or HA estates also on their doorstep.
How to improve comps that are not good? I suspect that depends on the factors that are an issue at any given comp. To provide a broader curriculum some comps near us team up: you can do 2 A levels at one and 1 at the other, 5 GCSE's and a BTEC at one and 1 at another, Be based in one school but do Latin at the other. MFLs spread across schools to give much wider choice. But the schools are within 7 mins walk of each other; not practical in a large rural area.
Money! It clearly talks in London schools, fight for good funding everywhere!
Make teachers' lives ones of joy and inspiration in unfolding their subject to their pupils, instead of bogging tnem down in paperwork and narrow prescriptive targets! Respect them, listen to them! (The government, that is).
I don't know, I am not an educationalist, but as the comprehensive model can and does work, let's prioritise fixing the schools that are not. If the schools are good the admissions panic may die down.