I personally don't think the Kobayashi Maru showed the beginning of the friendship, that was their first meeting where their different approaches started them off on the wrong foot. That animosity developed across the early stages of the film with Pike, when Kirk was made second in command.
It then reached a height with the bridge fight after Vulcan was lost (with a little help from Spock Prime). So we have the initial tension leading to the flare-up.
Then post-flare up, once Kirk is back and once Spock has worked through his emotions with Sarek, he returns to duty and he and Kirk must work together to board the Narada and save the day. This is where they begin to set aside their differences and use their respective talents in conjunction with each other.
Post flare-up they are now developing a trust and respect of each other. They are not friends in the TOS sense, but the development line is there.
'If the Apocalypse starts, beep me!' - Buffy Summers
'The sky's the limit.....' Jean-Luc Picard, 'All Good Things'
courtesy of Saquist