There also needs to be some way of determining diminished mental capacity and removing someone who is no longer capable of serving.
Such a mechanism exists at the Federal Level for the office of President.
From the 25th Amendment to the Constitution of the United States:
Section 4. Whenever the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmit to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office, the Vice President shall immediately assume the powers and duties of the office as Acting President.
Thereafter, when the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that no inability exists, he shall resume the powers and duties of his office unless the Vice President and a majority of either the principal officers of the executive department or of such other body as Congress may by law provide, transmit within four days to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office. Thereupon Congress shall decide the issue, assembling within forty-eight hours for that purpose if not in session. If the Congress, within twenty-one days after receipt of the latter written declaration, or, if Congress is not in session, within twenty-one days after Congress is required to assemble, determines by two-thirds vote of both Houses that the President is unable to discharge the powers and duties of his office, the Vice President shall continue to discharge the same as Acting President; otherwise, the President shall resume the powers and duties of his office.
The New York Consitution provides a similar provision regarding the Governor (Article IV, Section 5)
In case the governor is impeached, is absent from the state or is otherwise unable to discharge the powers and duties of the office of governor, the lieutenant-governor shall act as governor until the inability shall cease or until the term of the governor shall expire.
I would imagine (but I can’t say that I am sure) that the method of making such determination is governed by law.
The Wolf