In federalist 69, Hamilton said presidential power is inferior "to that of the governor of New York." The governor could cast a vote to appoint a judge. And in all caps, Hamilton wrote, "In the national government, if the Senate should be divided, no appointment could be made." The founding fathers realized the ability of the states, via the Senate, to check a nomination was one of the constitution's attractions. They did not want a president with kingly powers of appointment. And New York ratified the constitution.
Wilson below, showing congress can control the supreme court
The Convention found the task too difficult for them, and they left the business as it stands, in the fullest confidence that no danger could possibly ensue, since the proceedings of the Supreme Court are to be regulated by the Congress, which is a faithful representation of the people; and the oppression of government is effectually barred, by declaring that in all criminal cases the trial by jury shall be preserved. [bold face mine]-- Speech of James Wilson. Pennsylvania, October 6, 1787.
This means the congress can halt the proceedings of the supreme court. I'm not sure if the congress can eliminate the supreme court because it's charter is in the constitution. However, pertaining to nominees, the constitution says the Senate can just ignore the nominee anyway. No need to make a statement on it.