Another point of comparison would be state lieutenant governors. At least one, the Texas one, has influence separate from the governor but I don't think any preside over any state legislative chamber. A few are elected separately from the governor.
The Texas Lt Gov is an interesting example of what a VP could do...
“The
Lieutenant Governor of Texas is the second-highest
executive office in the government of
Texas, a state in the
U.S. It is the second most powerful post in Texas government because its occupant controls the work of the
Texas Senate and controls the budgeting process as a leader of the Legislative Budget Board.
Under the provisions of the
Texas Constitution, the Lieutenant Governor is President of the
Texas Senate. Unlike with most other states' senates and the U.S. Senate, the Lieutenant Governor regularly exercises this function rather than delegating it to the
president pro temporeor a
Majority Leader. By the rules of the Senate, the Lieutenant Governor establishes all special and standing committees, appoints all chairpersons and members, and assigns all Senate legislation to the committee of his choice. The Lieutenant Governor decides all questions of
parliamentary procedure in the Senate. He or she also has broad discretion in following Senate procedural rules.
The Lieutenant Governor is an
ex officiomember of several statutory bodies. These include the Legislative Budget Board, the Legislative Council, the Legislative Audit Committee, the Legislative Board and Legislative Council, which have considerable sway over state programs, the budget and policy. The Lieutenant Governor is also a member of the Legislative Redistricting Board (together with the Speaker of the House, Attorney General, Comptroller, and Land Commissioner), which is charged with adopting a redistricting plan for the Texas House of Representatives, Texas Senate, or U.S. House of Representatives after the decennial census if the Legislature fails to do so.
In the case of a vacancy in the Lieutenant Governor's office, the Senate elects one of its members to act as President of the Senate until the next statewide office election, in effect becoming the Lieutenant Governor. A Senator elected as presiding officer in this way retains his district seat and the voting privileges entailed with his Senate election. The Lieutenant Governor is sworn-in on the third Tuesday every four years, the same as the Governor.”
https://en.m.wikipedia.org/wiki/Lieutenant_Governor_of_Texas