While characters of all three lawful alignments could be autocratic rulers, only Lawful Good is hindered by morals, while both LN and LE put the state first, whether for law - order - tradition in and of itself or for his own sake respectively.
This is not true. Lawful Neutral
could put the state first, but Lawful Evil would most often put himself first. The fact that he may or may not be the state is entirely tangential to that. Selfishness (and acting upon it) is what ultimately defines Evil more than anything else.
Your stated dichotomy is still false. Good and Evil has literally nothing to do with the state. In the same way someone that is Lawful Good may oppose (or seek to change) an unjust state, someone that is Lawful Evil can and will oppose (or seek to change) a state that does not benefit him. Exactly how they do this is extremely debatable. Lawful Neutral, however, would generally have very little interest in whether the state is just or not and whether it benefits him personally or not, because to him, the being of the state is more important, even though he (like most people) would likely prefer to live in a just society that benefits him himself.
It is also a mistake to think that most people do this consciously. A Lawful Evil monarch would likely argue that while his system is imperfect, he is the one that is most fit to rule, for the good of all (and he might not even be wrong). A Lawful Good monarch could likewise argue the same, and maybe even be correct. A Lawful Neutral monarch could easily say that the existence of the state itself, while imperfect, is far preferable to anarchy, and benefits everyone.
All might argue with the best of intentions for all, but it will
not affect their alignment in terms of Good and Evil.