Because according to Harry and McCoy it's very existence is a threat to the WC, either as a scapegoat for all the BC activity or simply as proof that the WC is not actually (and cannot realistically be ) the end-all/Be-all Authority on Mortal Magic that they claim. Keep in mind that the BC is in the same boat, which Langrty has decided is too dangerous to even admit it exists. The existence of the Grey Council is in itself a big threat to the WC, but that is only the case because it is not officially sanctioned. If it were official and just secret (like the Blackstaff position, for example) it would not pose that danger; it onl;y becomes needed because Langtry is insisting on the policy of Denial:
I agree to all that and as you stated Langtry is insisting on the policy of Denial. My argument was that he guesses/is fairly certain about the existence of these two groups but has no more factual information other than their existence and combative nature.
I think it would be easy enough for the WC to deny until they are forced in to action against one side or the other. Both Groups are trying to be secretive why not let them be secretive until one is forced in to the open.
I also think that the one argument that you mention about being the supreme authority of mortal magic holds the most weight. That one brings Wizard pride in to it and the WC would lose face if it had a secret organization running around with in it.