What???
I have to disagree. I always got a feeling of close friendship between Spock and Kirk, and I got it looooooooooong before there were such things as Trek movies. Always. The most obvious sign (aside from Spock's comments that Kirk is his friend and Kirk that Spock is his, something which they both mention more than once during TOS) is that Kirk is the only person Spock calls by anything other than title and name, and he wouldn't do that if the friendship weren't close and mutual. The friendship between Kirk and McCoy was more obvious, but that's because McCoy is an out-there-in-the-open kind of guy, whereas Spock definitely is not.
What I never did really get, though, was much of a feeling of friendship between McCoy and Spock. I think I was supposed to - I think that's what all that TOS teasing was supposed to indicate - but it didn't work for me.
C'mon people. There's numerous examples of Kirk/Spockiness. They heart each other.
In Whom Gods Destroy, Spock of all people admits Kirk is his brother.
And who can forget Spock's famous smile in Amok Time? I mean, you think he was just concerned about being court martialed?
And then there's the Naked Time. "Jim when I feel friendship for you I feel ashamed" or something.
And then there's all the clear innuendo about the depth of their feeling for one another in all those other episodes that I won't even mention here.
I really don't get how anyone could get any other impression than that Spock and Kirk were very close friends. It's obvious throughout TOS.
In addition to this: "The Devil in the Dark" is one of the best examples: Spock is advocating the stance that they should not - and don't have the right to - hurt the Horta, as you would expect of him. But the moment he realizes Kirk's life is in danger, he forgets all about it and starts yelling at "Jim" to kill it before it hurts him.
"Amok Time", as already mentioned... "The Naked Time"...
Kirk considers Spock and Bones his two closest friends (I can't remember which episode it was - maybe "Operation: Annihilate"?), Spock calls Kirk his friend many times; and after the mind meld with Kollos, Kollospock calls Kirk "captain and best friend of many years" (that's the same scene in which he recites Byron's 'She Walks In Beauty' to Uhura

).
In the same scene, he refers to Bones as "another long time acquaintance", which reflects Spock's uncertainty over whether he and Bones are friends (which a bit more realistic maybe than Spock calling Bones his friend in "Amok Time"). I don't think it was inferred that they were friends - in one episode (again, I can't remember which one) as Spock and Bones are arguing, someone asks Kirk "Are they enemies?" and he answers "I think they're not sure".
I don't understand the idea about a romantic relationship rivaling or interfering with a friendship, or vice versa. It only happens when someone's partner is possessive, overbearing and jealous (not in the sense of sexual jealousy - but in the sense of being jealous over anyone and anything their lover is close to, which is really sick). There are people like that, but Uhura certainly doesn't seem like one of them.
Frankly, I always hated the way TOS sidelined the women - but what I resent even more is that some people see it as an ideal of sorts - male friendships (or more, if you are into slash) without any nasty females to spoil it all... how do you say it? "Bros before hoes", is it?
