I think Kirk would have tried to save any member of his crew and the bit about Spock leaving Kirk behind if their positions were reversed seems to show how tenuous their relationship still was. Moreover why wouldn't Kirk want, potentially at least, the best first officer in the fleet?
Kirk likely wouldn't have put any other member of his crew in that position because he doesn't trust them like he trusts Spock. Spock leaving Kirk behind says nothing about how he feels about him, logic would dictate he do his duty and follow the Prime Directive.
This Spock is a decade younger than the Spock we see in TOS, I think it would be a reach to call him the best first officer in the Fleet just yet. Enterprise is likely his first posting as an XO based on his age.