I don't think this will happen. Because it is predicated on accepting that there are some good schools and some bad schools and it's ok for some children to have to go to a bad school and all we need is a fair system for working out which kids to condemn to going to a bad school.
If you accept that, then you are right there should be a lottery. Rather than it being across a whole LA it would be better to cluster 2-4 schools that are nearby to each other with a range of quality levels, and have a single catchment area for the cluster, with random allocation within that cluster.
However I don't want that and I don't accept that.
I want every school that is undersubscribed and has a reputation for being a terrible place to be allocated, and which sharp-elbowred parents will pay through the nose for private, move house, jump through a multitude of other hoops and go through the Appeals process to avoid, to be given a huge boost of the personnel, resources and funding they need to revolutionise, reinvent and metamorphose into an amazing school that families are scrabbling for places at.
There really shouldn't be any mediocre schools, but if there are they should be the ones in the naice, safe, leafy suburb areas which are perfectly adequate and do a reasonable job getting a prescribed level of knowledge into the brains of their cohorts of reasonably able, reasonably compliant children from naice families who are switched on to and engaged with education. But that the really aspirational families will nevertheless also apply to the innovative and exciting inner-city school that they aren't in-catchment for and which was regarded as awful 8 years ago - because their offering is something so intrinsically desirable.
Your lottery system embeds the culture that there must be winners and losers. I want instead to make every possible outcome a prize worth having. And given the enormous costs that would be intrinsic in the chaotic quadrupling of rush hour traffic if such a lottery was instituted, I don't think my plan would actually cost much more.