Public school blue regularly fit for sport (5,6)

I believe the answer is:

rugby league

I'm a little stuck... Click here to teach me more about this clue!

'sport' is the definition.
Both the definition and answer are singular nouns.
Maybe there's a link between them I don't understand?

'public school blue regularly fit' is the wordplay.
'public school' becomes 'rugby' (I've seen this before).
'regularly' indicates alternate letters (regularly take one letter, leave next etc.).
'fit' becomes 'ague' (ague is a fit of fever).
The alternating letters of 'blue' are 'le'.
'rugby'+'le'+'ague'='RUGBY LEAGUE'

'for' is the link.

Can you help me to learn more?

(Other definitions for rugby league that I've seen before include "In professional game" , "13-a-side game" , "Thirteen-a-side game" , "Football game for 13 players" , "game for twenty-six players" .)

I've seen this clue in The Times.

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