"There is literally only one school that I can get into and it doesn't seem fair that I have no choice. "
The thing is, school choice is, in very many areas (not just London) a myth - and it is a very painful myth because the illusion of choice makes people unhappy with the only school they have any possibility of getting a place at. They feel that 'everyone else has a choice' and make statements like the one quoted above.
OP, IME it is quite uncommon to have a genuine choice of schools. In the rural area I used to live in, everyone's only option was the village school. Schools in neighbouring villages were filled with children from their own community, so a child from a different village had no chance of getting in. Similarly at secondary, the rural secondaries had non-overlapping areas in practice even if no catchments in theory - if you lived in villages A, B or C you went to school X, in villages D, E or F you were closer to school Y so that was your only available choice.
Equally in the large town I now live in, I can 'choose' to send my children to my nearest primary and secondary.... or have no school place within the town.
Parents where I work equally have no choice - we fill up from our catchment, neighboruring schools fill up from their catchments [we do have formal catchments still, though being in the catchment does not guarantee a place] so we have no space for children from other catchments.
Think of it not as a 'choice', but as an ability for you to state a preference between schools IF they have a place for you. There are very many other parents who do not have a genuine choice through having several schools where they have a genuine chance of a place.