But there's nothing.
There's nothing to indicate when or why they became friends in "Where No Man..." either, but we are clued in pretty quickly that they are friends. They were together for roughly a year in "Amok Time" when Kirk decides to risk his career and his one true love (commanding a starship) to take Spock to Vulcan. The characters may have well had a similar shaky early relationship in TOS that we simply weren't privy too.
You say there's nothing there that indicates a growing friendship and I disagree. There's Spock still wanting to serve with Kirk even after their early difficulties in Star Trek 2009
, there's Kirk risking his command and saving Spock on Niburu, there's the excitement as they walk to Pike's office when discussing the possible five-year mission, there's Kirk still wanting Spock as his XO once he regained command.
Damning two films because they have different time limitations to tell a story vs. seventy-nine live-action episodes, twenty-two animated episodes and six movies seems incredibly silly to me.