Shooting@pigeons, a 'local' school can easily become very exclusive due to demand. In that case people who live close by can be excluded from it and commute a long distance.
Stopping abuse of the system by the people in the OP's article only widens the catchment by a very small distance. It doesn't change the fact that people are going to say 'but it's my local school' when the nearest school is a good one (and distance isn't really the issue here - there will be other schools in near distance, we are talking about London here), and desperately look for something else when it's a bad one.
According to the DFE website there are 19 primary schools within a mile of Eleanor Palmer.
The stats are here for Camden:
www.camden.gov.uk/ccm/cms-service/download/asset?asset_id=2963374
On page 67
The most oversubscribed (the 'good' schools) were:
Emmanuel - 170 applications for 15 places (half the intake, the other half is on faith)
Fitzjohn's - 339 applications for 30 places
Fleet - 273 applicants for 30 places
Eleanor Palmer - 235 applicants for 30 places
Christ Church Hampstead - 174 applicants for 27 places
And the least (the 'bad' schools):
Netley - 71 for 60 places
St Alban's - 36 for 30 places
Argyle - 79 for 60 places
Carlton - 94 for 60 places
Brecknock - 99 for 60 places
Becoming a 'good' school is mostly down to random chance, albeit that it is determined by your intake. The most oversubscribed places, which are all single-form entry, have a big advantage in admissions because by taking in only one form it's much easier to attract 30 middle class pupils than 60.
Conversely, the two-form-entry schools are far more likely to be undesirable, because the microclimate of middle class applicants is far less likely to arise by chance (when you have only 30 pupils a year, it's possible to have a good year, some good teaching, some good SAT results, and then wham! you are at the top of the league tables and everyone is applying) when you are taking 60 kids in each year.
Apparently there might be an overall shortage of primary school places in Camden as a whole, but then if you read the comments here where numerous people rejected Carlton:
www.kentishtowner.co.uk/2012/07/09/why-it-matters-camdens-hidden-school-places-black-hole/
it seems that a lot of people that claim to have no local school actually did have a local school (Carlton), it just wasn't 'good' enough for them, and so they took whatever steps they felt necessary to find somewhere better, be it go private or whatever else.
'Boo hoo hoo my child can't go to the local school' is very rarely the issue.
It's more 'boo hoo hoo my child can't go to the best school in the area and I won't settle for anything else'.