Spock's death is the climax of the movie, and we learn of and see his deep friendship with Kirk from the outset of the movie. Peter Preston was a nobody who lived and died soley to emphasize how ruthless Khan was.Wrath of Khan works perfectly as a standalone movie, just as Into Darkness does. You don't have to have seen all 79 episodes to apreciate Kirk and Spock's friendship in WoK, just as you don't to understand Spock's realization of friendship in ID.
I don't believe this to be true at all. The death scene in TWoK is iconic because the relationship they've culivated over the years. TWoK is good enough to be a stand alone movie because all the facts you need to understand it are there. But take someone that doesn't know trek, or hasn't seen TOS and their reaction to Spock's death is different. Did the whole theater get all emotional when Scotty carried that dead trainee up to the bridge? Not even close to the emotion Spocks death envoked. Why? because there is a history there.
Likewise, we see Spock and Kirk interacting throughout Into Darkness. They bicker (as many friends do), but they like each other, although Spock doesn't quite realize what he has until he loses it.
Kirk and Spock's prior adventures together in TOS do add to the WoK death scene, in the same way that knowledge of Wrath of Khan (and the entirety of Trek) adds layers to Into Darkness (i.e. the death scene being a twisted alternate universe version of the same event) but they're not "required reading", so to speak.
They bicker like an old married couple.When you click with someone? Yes it is.
And what have you seen in nuTrek that makes you think they've "clicked"? ... from Spock's perspective?
I would submit it's because you know of their relationship in hTrek.
And as I said above, Spock doesn't realize what a friend he has in Kirk until he loses him. When Kirk is dying, he realizes what Kirk felt when Spock, his friend was about to die on Nibiru. But unlike Kirk earlier, he's utterly helpless to do anything about it.