Should there be any mechanic that lets you speed up an ally's cooldowns?
Soul Shroud, an old item that has long since been removed, was the only item ever made that let players boost an ally's CDR. No one ever got it because it was a crap item for its cost, and Riot felt CDR was something a player should spec for themselves, not rely on an ally for. Also, Riot hates auras, so there's that.
But does that mean there can't be any mechanics that reduce an ally's cooldowns? I sincerely hope not.