Have you ever played a tournament in this game?
They cry "Die you bastard" in tournaments, so tournaments are not sparring or mock combat, they are high seriously.
And in a real battle your clan members have no risk of dying, because of the 100% reduced Death option in the campaign options.
i would suggest they turn off blood decals in a tournament so it does not look like they cause bleeding wounds on each other, you and your kin and everyone else will always leave the tournament with the same hp as when they entered it. but beside that, there really is no reason - except an option for more safe one on one sparring - for why your family should not be allowed to fight each other in a tournament. its safer than entering a real battle. if you dont think that, then the argument that blood happens or taunts happen does no matter either, because they dont die in tournaments, regardless of your settings on death
another possible option is to be able to tick on and off for individual clan members whether you want them to join real battles and whether you want them to join tournaments, so you have full control over what kind of battles they participate in. so, if you have a little 18 year old son or daughter with all their focus points in medicine, scouting, engineering, and so on, then you can keep them out of every battle, but still have them participate in your army with other useful tasks. or, if you have a son or daughter who has all their focus points in one or more fighting skills, think of them as clear prodigies in fights. so their caretakers could see it as a benefit for them to enter tournaments. but again, this is your choice on whether you want them to or not
being able to ask anyone to spar together alone in the arena could disable taunts and blood decals in that fight, and once someone is knocked out, they can fall on their butt and then get up again instead of being knocked out completely, so there is no doubt it was just a training session, not an aggressive fight
i think the arena itself could use different taunts. knocking one guy out and yelling "look at them run!" and stuff like that, they just havent gotten around to record arena specific lines is my guess
i also want to add that mock combat takes place in the wild between predator siblings, and things get rough in those play fights, too, and can cause injuries. but its necessary for them to fight like this among each other to prepare for dangerous hunts when they grow up. it is the same idea in medieval times, you have to toughen up if you are going to enter battles. endless wars are unforgiving to those not prepared, even if noble and equipped in full steel. its better to receive injures that can heal and be a step closer to being prepared, than receive a crippling injury during a real battle that leaves you debilitated and frail for the rest of your life, or ending up death. i know that becoming crippled is not part of the game, but from a role playing mindset, its engaging to pretend that the risk is there. they might implement it later on, too. but death can happen if you enabled that, that is how i personally prefer it