It just says that the Senate should advise and consent on these nominations. It gives no further guidelines. So what should be the defined role of the Senate here? I'm not going to research the entire history of this issues. In all my readings about the constitutional convention I don't recall this specific issue coming up. But in having a framework for what the different factions at the constitutional convention wanted, I think we can come close to figuring out what they meant the passage to mean.
Many of the founders were worried about giving the executive too much power. And there are various places in the constitution where you can see this. So I would argue that because this isn't an instance in which they spelled out explicit details, we should assume that the executive should be given some latitude as to how much power it gets in appointing people to cabinet seats. If they wanted congress to have the power to appoint cabinet heads they simply would have given congress that power. But they gave it to the executive.
So even before we get to the text we are implicitly deferring power to the executive. When we read the text it becomes even more clear. In fact, the relevant power is provided in Article 2, which spells out executive powers:
He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
The text explicitly gives the executive the power. But where it gets complicated is "with the Advice and Consent of the Senate". So the Senate has some power within this process. The president doesn't simply get whomever it chooses. There is at least a very minimal check on this power. The question then becomes how much of a check? For most of our history we have been operating under the assumption that there shouldn't be much check and the president should get who it wants.
I don't think that's too far off the intent of the founders. I think the most important thing for the Senate is to make sure the nominee is competent and not too crazy. If a person meets those qualifications I think they should be appointed. Competent should mean having some experience and/or knowledge about the agency they are going to lead. And not too crazy should mean that you can have outside the norm views on the policies you are going to have an affect on. But you can't favor crazy stuff. For Secretary of Defense, that would mean something like favoring preemptive nuclear attacks or torture; stuff that is just way over the top. You can't make the threshold simply disagreeing about a particular policy or worldview; say Israeli settlements or being a realist vs liberal. Otherwise no one would ever get appointed, which seems to be the direction the Hagel situation is heading toward.
So the president should largely get who it wants as long as the nominee is qualified and isn't too far off the deep end. The latter requirement is a bit difficult to quantify. But I think it's sufficient to handle that on a case by case basis. And Hagel doesn't seem to hold any really crazy views. In fact, he's so mainstream that it makes Republican opposition to him look nuts. I'm not sure what their end game is considering they have 4 more years of a Dem as president. But eventually they are going to have to swallow the fact that someone will head the DOD that doesn't agree with every single view they do, while also being pretty qualified and not too crazy. As the founders and logic suggests, if they don't like this situation they should win more elections. That and Harry Reid and the Dems should do something about the filibuster, which is making the Senate an absolute joke.