Solomon was king. In those days a king’s responsibility included establishing and maintaining justice. The king was the highest court of law.
The king’s responsibility also included maintaining and defending the country. They were to establish and organize the fighting men.
The king was not responsible for maintaining the priesthood, for priests were established by birth family, just as the king was established by birth (if the king loved the Lord and served him). The king also did not tell the priests how to offer sacrifices and what to do for that was established in the law of the Lord. However, they could organize the priesthood as David and Solomon did.
David had left two unfinished responsibilities to Solomon. One included the army ranks, and one included the priesthood. Joab has stuck two important men down without David’s knowledge and even against his wishes and intent. David had told Solomon to deal with Joab. Solomon took this event to have Joad put to death.
The Lord had told Eli, the high priest what would happen to his household because he did not deal with his son’s defilement of the priesthood. That would have been around the time of David’s birth. Solomon now banished Eli’s grandson, Abiathar from serving as a high priest.
Solomon used the event concerning Adonijah to enact the Lord’s word. The Lord’s timing of justice is not often as I would like. Justice often takes a long time. Perhaps because God wants to give time to the person to repent. Perhaps there is another reason. Whatever the reason, I am keenly aware that the Lord is also slow to be angry with me and his justice towards me is merciful. The Lord is full of grace and mercy to me just as he is to everyone.