Currently, if one doesn’t own Reforged but does have an original Reign of Chaos game key linked to their Battlenet account, the (beta) BDA doesn’t detect it. Instead, when trying to install, the owners of the original WC3 are met by a “Buy Now” button instead of an “Install” button.
It’s circumventable, but it’s not obvious and has to be explained to the vast majority of players.