There are some very odd statements and assumptions in the article:
A small number of pupils who applied for and sat more than one test will be included in the figures.
As we well know, there is more than a "small number" of pupils who take more than one test. The crossover in some large 11+ areas such as Berks/Surrey, Bucks/Berks, Herts/Barnet, Kent/Medway/Bromley is considerable, with some children taking 4 or more different tests, and many taking two as a matter of course.
They do not include children who have narrowly missed a place at "superselective" schools, which only take the top performers and do not have a pass mark.
In which case, which schools does it include? I find it scarcely credible that 15,000 qualified pupils are turned away from grammars on distance rules alone. In Bucks, where there is a flat pass mark, there are sufficient places for every qualified child, albeit those places may not necessarily be easily accessible and parents may have to turn them down for that reason. It will still amount to no more than a handful of children though.
If a similar number of eligible pupils are being rejected from the other 108 grammars in the country, it would mean that nearly 20 more establishments would be needed to meet the demand for places from about 15,000 extra students.
This is the problem - the article is based on an extrapolation that goes waaaay too far.