Never in a million years would I believe Kirk would think, "I need to redeem myself as commander of this vessel by sacrificing myself". Even Deanna Troi was taught to sacrifice a friend to save the ship, not sacrifice the ship's captain.
Isn't that exactly what he was offering to do in The Wrath of Khan?
KIRK: Khan, ...if it's me you want, I'll have myself beamed aboard. Spare my crew!