I grew up in the US, in a suburb of a major city. You went to your nearest primary.
You're very likely mistaken about one major item. You probably went to your "hard catchment" primary. If you lived near the edge of your catchment, you might have been nearer the primary in the next catchment, but what mattered was which side of the catchment line you were on, not the distance.
The distinction, which other posters have pointed out, is that London doesn't really have "hard catchments". What people very loosely use as "catchment" is the distance within which children normally can get a place at a school. No guarantees if a new apartment block goes up or in a year with an unusual number of siblings, etc.
You are correct that, in most US suburbs, a school is required to accommodate all students living within its catchment lines, expanding capacity if necessary and that this is a major difference to the UK system where schools have fixed capacity and students need to take up the adjust.
You failed to mention that one consequence of this is that students have essentially no choice in their public (state) schools. They are usually guaranteed a place in the catchment school corresponding to their address, but they cannot choose to attend a school in another catchment even if, as I mentioned above, that happens to be the nearer... or just much better. I think very few Mumsnetters would be willing to support a system which eliminated parents' ability to choose their children's schools.
One major problem with the UK system, at least as it pertains to London, is that it prioritizes giving as many children their first choice as possible without taking into account the cost to children who don't get their first choice. To illustrate, imagine that Child A lives 600m from School A, Child B lives 400m on the other side of School A, and School B is 500m beyond Child B's home. So, in a straight line:
Child A ... 600m ... School A ... 400m ... Child B ... 500m ... School B
If Child A and Child B both prefer School A, Child B gets the place and is 100m better off than if they had to attend School B, with a 400m walk rather than a 500m walk. The problem occurs if School B is also Child A's second nearest school. Now Child A is 900m worse off than if they'd been allocated the place at School A, with a 1500m walk rather than a 600m walk. That's how "black holes" form in London. Too many Schools A fill up with Children B, and some unfortunate Children A are allocated Schools B.