That's not a weakness. He knows that he's best equipped to captain a starship, which does not run on programming, but on living thought, a soul, and the idea of purpose, even destiny. M5 proved it was not up to that task, and was ill-prepared for even the most common sense tasks when dealing with personnel selection, the war games, etc. The point of the episode was to expose Daystroms failings as a human being; while he--as noted by Kirk--believed murder was against the laws of God and thus imprinted that in the M5, but he also overload it with conflicting, clinical directives and ideas that do not work in reality, hence Kirk's rational rejection of the use of the M5 to run ships.
It was his ship. Captains are not transient employees who have no deeper connection to / investment in the use of a ship. Its an extension of one of the reasons captains were out in the "final frontier" in the first place. One drives the other, so they are so joined together, that Kirk knows it is his ship.
