What did Pike tell him after the alleged maturity he learned from the Nero incident?
In this movie, he shows not repentance for lying about breaking orders. He undertook the mission to find "Harrison" without questioning the morality of it -- much less the fact that firing on Q'onoS (not "Kronos"!) would be an act of war. When Scotty resigns, Kirk gives Chekov, who's not an engineer the job of chief engineer instead of the actual assistant chief engineer, a serious mistake, given how they needed engineering to be run by a qualified officer when the ship shut down. He would've killed his entire crew, except that Scotty saved his behind. He willingly partners with Khan against his better experienced first officer's advice, a war criminal (although not the tyrant that Spock erroneously describes him as), which leads to Khan's kamikaze strike on San Francisco, which had to have killed countless people.
How has he showed that he's ready for the big chair? Just because he wants to do the right thing and comes through on that has no bearing on his qualifications to command the Federation flagship.