policywonk - yes, I do see your point about expensive houses in catchment areas for good schools. When we were looking to move, we saw such an area - half a million pound homes surrounding a great school, in fact it did look like it was a deliberate decision to build the houses around the school in this way on the part of the town planners, there were only about 10 or affordable houses there (we wouldn't have been able to afford even those). That was the exception though. I understand most local authorities have to build a certain percentage social housing and affordable homes in an area.
The place where we now live has a mix of different types of housing, including council housing. It's a small town with three primaries, and we're very happy with the local school because we chose it based on the area. What we did find, was that there are very many very good primary schools everywhere, and we had a wide range of areas to look in.
A lottery, on the face of it, seems like a fair thing, to even out the playing field, but not being able to go to a school on your doorstep because the lottery deemed otherwise is crazy.