Round here (SE London) one of the issues seems to be that there are a lot of schools within a very small area i.e. 4 primaries within 5 streets (not main roads, residential streets). Each of the schools is very different although two of them share an Executive Head. One is open plan, another is a redesigned Victorian building across 3 storeys and another is a Victorian site but with three buildings and all teaching spaces are ground floor. This means that parents may well have quite a strong preference for one type/ethos of school over another and yet all the schools will be within 1000m of an individual's home.
I do not like the idea of a lottery as I do think siblings should be placed at schools together (if the parents want and if they live the same distance from the school as when the first child started-see below). It would have the potential to make life very complicated especially with regard to forming local friendships, being able to walk home from school (at a later age, not 4!) and all sorts of other things.
At present the system isn't that bad although it feels horrendous if your 4-year-old is unfortunate enough to be offered a school 2.8 miles away, in the opposite direction from the school you teach in, is only accessible via two different buses, has no breakfast/after-school club and there is no local (to home) childminder who will take on a child who needs to be delivered to a school so 'far' away. This is what happened to us and, had I not been on maternity leave at the time, either myself or my DH would have had to seriously consider leaving our teaching jobs in order to get DD1 to school. What actually happened was that I declined the place and accepted the one offered 5+ miles away, opposite the school DH teaches at; it wouldn't have been ideal but at least it would have meant we could both carry on teaching.
The Admission Code was introduced to make things fairer for everyone and whilst I believe that's the right thing to do, there's a part of me that would like some sort of priority criteria to be applied for Reception-aged offspring of teachers! Realistically I don't know how this could be applied; perhaps it should only be applicable if a school is at least 2-form entry and the child lives within a mile of the school.
The other change I would make would be to make sibling priority applicable only if the family home was the same distance (or closer to) from the school as when the first child was offered a place. It's frustrating to know that your own child has missed out on a place because X, Y and Z have all got places merely because they have siblings and that they now live much further from the school. Again, this wouldn't be easy to administer and something would have to protect those in social housing/those who have to move due to exceptional circumstances.