The authority of the king

Hello Rabbi
Following the column about the death penalty for terrorists and the discussion surrounding it, I wanted to ask your opinion about the king’s authority.
After the division of the king and the apparent separation of powers, it is clear that the king’s authority derives from the law. It is true that the king and the secular authority are the ones who appoint the judges, but if our starting point is religious (the commandment to appoint a king), then it is clear that the Sanhedrin will naturally inherit the king’s place and will be able to amend halachic regulations (after all, the commandment to appoint a king allows one to recite the Torah blessing).
One can add to this the argument that if the authority of the written and oral Torah is Torah/prophetic, then all the commandments are ultimately religious.
In short, if the palate fills the gaps in religious law, then its function is also religious.
(I do remember that there is a midrash that sees Moses as also functioning as a king, but that seems more legendary than halakhic.)
Thank you very much.