You have to think if faith schools (other than the new Ines of the Blair years) as church schools that are currently operating in cooperation with the state system. The state doesn't and never had owned them.
Even in the days when we thought we were rich, no government has attempted to buy out the owners. It would be a very expensive policy in itself, plus the capital cost to the government of schools would increase, and quite a number of church/community funded improvement projects would simply never happen.
The role of covert selection isn't as clear cut. If you live in a diverse community, you will know that many church schools have an intake which is predominantly made up from communities of recent immigrants because churchgoing isn't just something done by middle class 'naice' people in leafy areas to get a "selective" school. The proportion of CofE schools is so great as to be formative of the Mormon. If you look at Catholic schools, they come out as typical in terms of proxy markers for deprivation (yes, there are leafy ones, just as there are leafy secular schools, neither is terribly representative of all RC or all secular), but are more diverse in terms of ethnicity.
One feature that interested me (though I haven't checked recently to see if it has remained the case) is that RC schools - a sample demographically matched to other schools - had the lowest rates of exclusion. I've never seen a plausible explanation for why that should be the case, other than pastoral care and ethos.