The problem is that the only constitutionally guaranteed power of the "President of the Senate" is a tie-breaking vote. Anything else comes from the Senate rules, which are set by the Senate itself, so the VP's powers in the Senate are dependent on the consent of a majority of the Senate. Pretty much the first time the VP is either a different party from the Senate majority, or the first time there's a major personality or policy conflict between the Senate and the VP in a non-partisan or one-party environment, the Senate is going to vote to change its rules and nerf the VP's powers as President of the Senate.