I haven’t studied too closely the rationale behind the VP being the president of the Senate. Was the intention for the second most popular man in the nation to act as a sort of Prime Minister to counterbalance the president?
No, his role is to cast a tie breaking vote in case of disputes between the States that resulted in ties in the Senate. Remember, the Senate was suppose to be made up of 2 men selected by each State to represent that State on federal issues in the Senate. There is really no relationship at all with the Executive branch.
The Senate was originally set up to balance powers of the states against the power of the Fed, while the House was to be the voice of the people. I think the VP was meant to be the federal voice in that body.