I'm in the US and just wanted to point out that they use lots of different systems here, not just 'are you in catchment'.
I'm in a small town and the only kids in our schools are those who have a parent living in the town limits (and most importantly paying property tax). So it is completely catchment based. If more houses are built they either build more capacity or move things around. Right now they're doing both. We do have spare capacity though - we don't have 30 kids per class, and when we start approaching that we build more capacity.
In the nearby city it is not done by catchment at all. There are two public schools and multiple elementary/middle schools - in many cases children are bused from one area to another, depending on which school their parents chose, and various other factors.
There are also various charter high schools, which allocate places by lottery to anyone within the state - for practical purposes you do need to live somewhere near enough to commute in daily, and you have to find your way in - no school buses. They do prioritise siblings.
There is also a state magnet boarding school to which anyone can apply, and they select by various criteria including SAT score, essay, interview, and whether they think the student will benefit from going there (so if you're coming from a terrible school district they're more likely to take you).
And this is just one state. Other cities/states allocate school places their own way.
Also it's not necessarily true that the richest towns have the best funded schools. In my state it's the schools in the poorest towns that are best funded - the poorest county spends $24,000 per student on education. I'm in a fairly well off county and in a very well off town, and we spent $13,000 per student. Our cost per student is pretty low by state standards, but our schools are consistently ranked high within the state and the country.