The real issue, in my mind, is not summer but the rest of the year; it seems odd to me that we don't have boarding schools for students that are 1 in 10,000, where they can learn about a high school course a week, or do things like this.

The impression I get from his discussion of the French & Russian elite schools is that the summer courses may be better considered part of the filter for admission into those sort of schools; specifically, footnote #150:

The network of Russian maths and physics schools initiated by Kolmogorov (one of the 20th Century’s leading mathematicians) educated many faculty members of western universities (and Grigori Perelman, who solved the Poincaré Conjecture in 2003). The experience of these schools is that 1) they must take great care in selecting pupils (e.g a summer school system); 2) they need complete freedom over the curriculum and testing (Kolmogorov had students learning set theory, non-Euclidean geometry, topology, and other subjects not in a usual school curriculum; the Novosibirsk school used Feynman’s famous Lectures as their textbooks); 3) ideally they are integrated in the life of a university with academics teaching maths and sciences. These schools gave me the idea to copy them in England.

This makes sense since cognitive testing has known measurement error, which while very low, is still problematic if you're trying to select the top 1% (or higher) of students* for these expensive schools (and because intelligence itself changes over time, so a single test means regression to the mean), and it may either spark enthusiasm for going to the advanced schools or test the child's attitudes toward it - not much point in enrolling a kid who doesn't want to be there, right?

* for example, apparently the Termite study missed out on two Nobelists because they were under the filter by some tiny amount on the test; if Terman had been able to afford a larger sample or had a slightly better test, his results would have been much more impressive, especially since Nobelists are one in millions.