The Vice President of the United States shall be President of the Senate, but shall have no vote, unless they be equally divided.
The Senate shall choose their other Officers. And also a President pro tempore, in the Absence of the Vice President, or when he shall exercise the Office of President of the United States.
My interpretation is there is intent the Vice President has a direct active participates presence in the political decisions of the Senate. It limits and restrict vote to being the tie breaker vote, but use of President pro tempo indicates a power to preside in the absence of the Vice President. Although duties of President pro tempore are essentially ceremonial, this does not mean the Vice President duties to preside in the Senate limits the Vice President to being a silent unseen officer and member of the Senate.