It is logically false to equate a failure to obtain unanimous consent to vote (or in the alternative, cloture) with a simple minority vote to reject the nominee. Under the first, the President is denied the appointment of officers of his choosing, and the denial is implemented by less than a simple majority of Senators. Under a rule of "to reject the nominee, you must vote on the nominee," a simple majority of Senators would consent to the officers the President has nominated. The difference to the President is between having, and not having the officer of his preference.
Senate procedure has no method to dispose of a treaty by not voting on it. Postponing a treaty vote indefinitely (effectively the same as passing it back to the President) takes 2/3rds, and does not dispose of it. It is easier to reject a treaty, with 1/3rd of the votes, than it is to postpone the vote indefinitely, which takes 2/3rds of the votes. And in the case of rejection, the dispostion flows from a vote on the treaty, not a procedural step that determines whether or not they vote on the treaty. A Senator can't hide behind unanimous consent or cloture to kill a treaty.
The presence of a protocol for postponing voting on a Treaty, other than unanimous consent or cloture, raises a question about whether a cloture motion is proper in the context of Nominations. Cloture abuse gives a 2/5th minority the power to "terminate the nomination." What a contrast with the 2/3rds supermajority required to scuttle voting on a treaty.
How does handling of a nomination tie in with handling of a treaty? Both involve powers granted to the President. The president has the power to negotiate treaties and submit them to the Senate for their advice and consent. The President also has the power to appoint judicial and executive officers of *his* choosing, with the advice and consent of the Senate.
By invoking the tool of cloture (to withhold voting) in the context of a nomination, the Senate has erected a supermajority hurdle of ITS OWN CHOOSING, using a Rule that did not exist until 1917. The hurdle in that rule has been amended on three occasions. If the Senate is free to erect and change the hurdle for nominees, what's to prevent the Senate from setting it at 2/3rds, or 3/4th, or 9/10ths? Or to prevent a SINGLE Senator from withholding the vote - exactly the way the Senate worked before the cloture rule? According to some Senators, the Constitution gives the Senate the power to make its own rules. They are right, but they over-reach when they apply their rulemaking power to nominations.
The Senate is free to make rules that affect only it. It cannot make rules that diminish the power of another branch. The Senate has a DUTY to advice and consent, confirm or reject, each nominee. It cannot consider and then refuse to vote on a treaty. It cannot consider a nominee, and then refuse to vote on the nominee. Even though the Constitution does not say the Senate has the duty, it could not conduct an impeachment trial and then refuse to render judgement. To do so would neutralize the impeachment power of the House.
In contrast, the power of each House to make its own rules clearly applies to matters PURELY internal to its workings. The limitation is in the language of the Constitution, "shall be the Judge of Elections, Returns and Qualifications of its own members ... may determine the Rules of its Proceedings, punish its Members for disorderly Behaviour, and, with the Concurrence of two thirds, expel a member."