You are being unfair to the LEA; in my experience they were always on the side of the parents in appeals and did everything in their power to make the oversubscribed schools take extra pupils. But they cannot remove existing pupils to make space for your grandchildren, and in an overcrowded class one extra child is the equivalent of five.
Unlike Academies, LEA schools are not allowed to create extra classrooms to accommodate rising rolls, hence the lack of places.
I am assuming that your eldest grandchild has a place at an Academy school, where none of the restrictions apply; they are allowed to create extra classes according to demand . These schools are increasingly run by larger and larger Trusts, whose motive is profit, (witness Wakefield City Academies; all 21 schools handed back because they are failing.) Not all Academies are run like this, but an increasing number are.
We seem to be going round in circles; when I was due to start school in the early fifties, and having lived in the catchment area for a good school from birth, my parents were told that I would be sent to a school some miles away, not on a bus route, because I was an only child, therefore no sibling connection. My mother thwarted this by alerting the LEA to the number of children given places by using grandparents' addresses, when they lived outside the catchment area. Still happens today.