Yes he will take it up but it could never be the next day. Handing him the keys to the cave and access to the suit does not instantly make him Batman. I agree Nolan's ending is Blake will inherit Bruce's Legacy. But it wass Bruce's story not Blake's. So he gave very little details on what Blake will do with it. He could have had a more defined explanation for things if he wanted. That is not to say his intention is not there. It is but in broad strokes not details.