Chess is certainly a game: a source of recreation; a pastime or diversion (as some dictionaries rather broadly define sport). Without doubt, it is a game which may promote mental development. But does it really require physical skill or exertion, the characteristic of most definitions of a sport - including the legal one which determines what the Sports Council may fund?
Therein lies the rub. Spitball would presumably count. Even Pooh sticks might stand a chance (which will cheer the Cambridge University team seeking a quarter-Blue on a par with Tiddleywinks). But the game of strategy is barred from lottery lucre, because it is insufficiently muscular.
Sport or not, chess is certainly worthy of financial support. But it ought to be possible to fund it properly without pretending it is something it is not - and in the process opening the doors to poker and pontoon.