It’s kind of hard to picture the TNG crew having a disagreement, however minor and somewhat jovial, over something political. More likely, the “correct” side would be obvious and the characters all on it. Maybe Worf would say something, but probably not.
Yep. Not only would it be true, but all right-thinking people in the society would agree. It was a progressive utopia.
One of the evocations of Spock’s words is owed to Plato, who discussed the idea of “the one” and “the few” and “the many” in discussing the Republic. Plato idealized a Republic in which all three forms were harmonized in a well-balanced whole governed by a philosopher-king, and he indicated that such a Republic would not last long and would result in a cycle of decay from tyranny to oligarchy to mob rule and then back to Republic, but although this was not likely the case.
Spock is not speaking in utilitarian terms, because he uses the crucial word “needs.” John Stuart Mill’s utilitarianism is principally about happiness, not needs. Happiness is subjective and needs are objective, and so, when Spock is speaking, he is speaking about a more fundamental idea of a polity – its self-preservation rather than its moral or psychological improvement – that in some ways, Spock is trying to evoke Plato’s idealism about societal essences, rather than a collection of constituent governing entities such as individuals.
Indeed, Spock’s words are meant to be an admonishment of Kirk, who had previously used the phrase earlier in the film, and so when Spock speaks Kirk’s words at Spock’s death, they evoke a wisdom of the philosopher-king (in this case also a man of action) who sacrifices himself for the good of the ship.
Spock is saying that the wisdom of “the needs of the many outweigh the needs of the few or the one” is itself the hallmark of the leadership of the one, and in this way, Spock imparts his wisdom to Kirk, for his benefit, so as to justify to Kirk himself, on moral philosophical grounds, the appropriateness of Kirk resuming his command as CAPTAIN of the Enterprise, and not an admiral training cadets on simulators. It is an admonition of Kirk’s abdication of his leadership, of the apparent loss of value to his aging life, and it is a valorization of the beauty and rightness of assuming command for selfless reasons when one is fit.
Whoa! Hea-vy! 
This was written in the 1st century by the Apostle John:
John 11:49-50 But one of them, Caiaphas, who was high priest that year, said to them, “You know nothing at all; you do not understand that it is expedient for you that one man should die for the people, and that the whole nation should not perish.”
It’s a vague echo also of Shelley’s Mask of Anarchy:
Rise like Lions after slumber
In unvanquishable number,
Shake your chains to earth like dew
Which in sleep had fallen on you -
Ye are many - they are few.
Nimoy was long active in the Jewish community, and could speak and read Yiddish. Consider that he first saw what became the famous Vulcan salute, “live long and prosper,” as a child at an Orthodox Jewish synagogue service in Boston. It’s understandable Nimoy would add other Jewish phrases into the script of Star Trek.
And yet most people instinctively agree with it, as shown by various versions of the Trolley Problem.
When it comes down to brass tacks, I think you would agree with it too–unless you think that it’s ok to kill five people in order to save one.
False dichotomy. One might, for instance, take the position that it is never OK to kill any number of people, to save any number of others, regardless of what the numbers are.
Just like that Famous Russian Epic of “Cinderella”…
or this Klingon joke
A Federation crewman arrives on Qo’Nos
He approaches a guard, and asks “Can I get to the Great Hall?”
The guard replies “If the bugs do not bite you…”
Most seem to be wandering off into the great beyond whereas the question was - What is Origin of “Needs of the Many…”
Then a mass-murderer on the loose shouldn’t be killed, to prevent more killings?
Conscientious military leaders sweat the number of their troops who must die to achieve an objective. I fear those who don’t so sweat. Peaceful resolutions would always be nice, sure. But the Trolley Problem looms, whether facing human or other threats to a group’s safety. How to PREVENT the necessity that some must die so others may live?
Yes, some would take the stance that killing the mass murderer is unacceptable, and would make you just another murderer.