You want a PRINCE in an absolute monarchy kingdom to be prosecuted by his family members for killing a traitor, who was actively threatening him and his father, the de-facto King. A traitor, to whom none of the people involved in this secret care. In the middle of the End of the World for these characters.
It would just be ridiculous in any aspect. It's Roshar, not Earth 2025. They have kingdoms, aristocracy and slavery. Even the best of them are just starting to question the lighteyes/darkeyes sistem.
Adolin being a prince is not plot armor, it's his place, established since book one. That's the reason Sadeas is after him. That's the reason he was at the Tower, betrayed by Sadeas. That's the reason the two of them have a history together.
Yeah, life isn't fair, people get away with things that would put others in prison. SA is about many things, but life being fair is not one of them. Taking responsibility for your actions doesn't involve "going willingly into a trial and facing the law" on Roshar.
Dalinar never faced a trial for his war crimes, he is only torchered by his conscience. Shallan killed both of her parents, Tin, Illai. Jasna goes around hiring assassins and killing people in the dark alleys. Amaram killed Kalladin's frends, Kalladin was involved in a plot to kill Ehlokar, Szeth killed a bunch of kings, including Gavilar, etc., etc., etc.
It's an epic fantasy book. People kill and don't get judged for that. Prosecuting Adolin for what everyone else is constantly doing would be the most stupid and unrelatable thing in SA ever.
You hate Adolin, we got it. You don't need to change your mind on this, you just can hate him without inventing plot armor and consistency problems that are not there. Adolin has a pretty big character arc for a character who wasn't in the main cast for most of the books.