Nlondondad, the argument set out by the original poster in that thread is flawed because it assumes the take up of the free school offers will be of a similar proportion to the other local schools.
However, as the free school had a separate application process, people will have only bothered to apply if they either a) were genuinely attracted by the proposed school's offering and/or b) felt uncertain about getting any of their higher preferences. They are therefore a self-selected group, and take-up is reasonably likely to be higher than predicted.
In contrast, many of the lower preferences on the pan-London form will be space fillers - people who put down 6 schools even though they're pretty confident of getting one of their top 3.
Of course take-up at the free school will also be dependent on the school showing good progress towards opening and communicating well with parents. People are naturally nervous of new schools, so take up is likely to be higher in the second year than in the first, as the school will have a track record.
For context, I think the school/dfe were probably being vocal about the application numbers to encourage others to get over their nerves and apply - no harm in that. And if their enthusiasm seems hyperbolic that may be because they were comparing numbers with other similarly sized free schools, rather than with established schools.