A strange question based on a distorted conception.
It is meaningless whether the minor is exempt or liable. Save the socio-legal babble for the house of babblers, where they like discussing all sorts of windbags about the legal-babbling distinction of the person being judged. It’s like asking whether Hitler was liable or exempt. The Torah is not interested in that chatter. What matters is what one does, not the social-legal-babbling-scholarly status.

“If a man has a stubborn and rebellious son, who does not obey the voice of his father or the voice of his mother, and they discipline him but he does not listen to them, then his father and his mother shall seize him and bring him out to the elders of his city and to the gate of his place. And they shall say to the elders of his city: This son of ours is stubborn and rebellious; he does not obey our voice; he is a glutton and a drunkard. Then all the men of his city shall stone him with stones and he shall die; so you shall remove the evil from your midst, and all Israel shall hear and fear.”

For far less than murder, a minor is put to death.

The Last Decisor??? Sabbath desecration, as if?? A minor is liable to stoning??? What is going on???

Dissenter, don’t be surprised; it seems that in general our master, The Last Decisor, is not bound by trifles like Mishnahs, Talmudic texts, and the tradition of the Sages. Fortunately, in some obscure book called the Torah of Moses, he still has some standing.