The first thing I’d like to say about boxing aliens is that human beings are likely mediocre fighters. So half of all alien species are stronger than us just like half of all alien species are weaker than us. For Captain Kirk, who would fight them all if he could, he probably shows the signs of the unscientific term ‘punch drunk’.
Of course Starfleet is more than possibly aware of Kirk’s condition which is likely a brain injury of some sort. They likely checked him out from top to bottom and deemed him able to still perform for Starfleet. And as for all his fighting on behalf of Starfleet, he was given the reward of being the youngest captain of a battle ready Starship.
They know his two weaknesses from the brain damage. The first one is just a speech tic that has only minor consequences. Kirk. Likes. To. Speak. In. One. Word. Sentences. This is annoying but hardly worth stripping the captaincy from Kirk. In fact stripping Kirk of anything might be seen as insensitive to minority groups. And the Federation has lots of minorities.
The second brain damage induced decisions are that Kirk frequently goes down to the planets himself. This can only seem fool hardy to those familiar with the military establishment.
Maybe Starfleet sees it as a way to get rid of Kirk in case his delightful brain damage starts getting worse. Maybe they like the way he continues the charade that humans don’t get brain damage despite repeated blows to the head. Maybe that’s one myth Starfleet wishes to see go on.
Regardless, Kirk’s good ol’ boxing worked for Starfleet. So why not let it continue?