Meaning of radical punchline [closed]

The term "radical" is probably being used in accordance with definition 2.2 in this entry of the Lexico online dictionary. This reads:

Characterized by independence of or departure from tradition; innovative or unorthodox.

In other words Rudman is challenging the conventional view that school is the best environment for learning mathematics. In fact he is using the observation that the ancient Greeks as a society excelled in mathematics without teaching it in school to argue the exact opposite.

The same dictionary includes this definition of "punchline"

The final phrase or sentence of a joke or story, providing the humour or some other crucial element.

Which indicates that a "punchline" need not, necessarily, be humourous but can be a succinct summation of an argued point.

This is an unusual use of "punchline" but is, I believe, what was intended.