For Kirk to say that Spock was the most human, despite rarely showing emotion, seems ridiculous.
Spock actually displays emotion frequently. His emotional displays are simply subtle and more nuanced than those shown by Kirk or McCoy or Scotty. Spock shows way more emotion than most people seem to remember.
Tuvok was cold, stone-faced, and condescending. Spock was stoic and intellectual, but he wasn't the emotionless automaton he's often remembered as.
Spock always has a logical justification for his actions, but in the end Kirk knew his friend well enough to know that a major part of Spock's motivation for his sacrifice was emotional - his friendship with Kirk and his concern for the safety of others.
A certain (paraphrased) Biblical quote seems appropriate: "No greater love has a man than this, that he would give his life to save his friends."
The points about Spock trying to better himself are very true.