I am not certain the VP has no power in the Senate, has one ever asserted their presidency of the Senate or just by custom he only reports for tie votes.
I suspect LBJ did or tried.
But what did he accomplish as VP?
The most the VP could have other than that tiebreaking vote is the power to lobby. But hell, EVERY former Senator has that.
Naa, I think someone like Pence would be a good choice. Good optics, Executive experience, Legislative experience, Respected enough on the hill to be a real asset to a president with zero government experience.