Name a classical monarchy, tyranny, or republic where the ruler does not have the right/ability to remove cabinet-level members of the government. Even in modern republics this is a standard thing. Pretty sure in modern monarchies most monarchs have the right as well, even if it's not practiced (due to possible protests against -them-).
The ruler should not have to call on the general membership to protest a person out of a position. The person could do an adequate enough job, but the purpose of a cabinet/council is historically to be a trusted group of advisers. i.e. a privy council.
In monarchies, the default title for the judge is Arch Priest. In ancient times, whenever the arch priest and the king were not the same person, the arch priest had the power to appoint the ruler (or crown him, otherwise) and was otherwise untouchable by the king. As for modern republics, supreme court judges are usually completely protected by law from the government in power.
I know I personally RPed along these lines as Arch Priest of Fwuvoghor, was loads of fun. Independent judges in conflicted realms are awesome, if you ask me.
Imo, the best way to deal with this was if the council powers were actually used, so that a weak ruler could not dismiss a strong judge, but where a strong ruler could dismiss a weak judge. As long as council powers are only modifiable following rebellions or anarchy.
