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Xeviat
I don't know how much is it balanced, but I would change the barbarian to be able to use one range per short rest. This means that you will be ignoring the number of ranges/day column of the barbarian table. Also, this change is flavorously reasonable: the barbarian will rage as the opportunity presents it to, but will be tired enough to be unable to rage again until it rests.
I know that this custom rage is stronger than the original at the first levels, but I think it balanced for 6th level barbarians.
An alternative would be to create a new barbarian path that reflects your vision of the barbarian. Here is a drafty one: (lack playtesting)
Path of the Insane Warrior
You can call this path as you please, I didn't come to a good name (nor to the features), but this path is intended for the barbarian that leaves it's emotions of madness, insanity, bloodthrist and fury take control of it's body and give it unthinkable destructive power. Also, it can enter rage quickly and often.
Bloodthirst
Starting when you choose this path at 3rd level, you becomes so insane while you are raging, that whatever damage you takes, makes you stronger. Whenever you take damage while you are raging, you received a bloodthrist dice, which is a d6. Whenever you hit a creature with a melee attack until the end of your next turn, you can add any amount of bloodthrist dices you have to the damage. You loose all unspent bloodthrist dices at the end of each of your turns.
Unstable Mind
Beginning at 6th level, the primal ferocity that represents your rage is the strongest feeling within you and you are so unstable that anything harmful can easily trigger your rage. Whenever you take damage while you are not raging, if that damage reduced your hit points to half your hit points maximum but didn't let you uncounscious, you can use your reaction to enter rage. entering rage this way doesn't count against the number of times you can voluntarily enter rage per day.