1) The LE person is E;
2) Hence, the LE person cares for nothing but his/her own self-interest;
3) Hence, the LE person acknowledges no other constraints on his/her action;
4) Hence, the LE person doesn't acknowledge the law as any sort of external constraint;
5) Hence, the LE person is NE;
6) Hence, LE is oxymoronic.
- numbers added for clarity
You go wrong at #2, and by failing to state 1.5 "The LE person is lawful", which would contradict #2.
Statement #2 is statement that the person is chaotic, and from there it follows that the person you've described is CE.
The correct order would look something like:
1) The LE person is E and L.
2) Hence, being lawful, the person believes that he has a duty to a higher power whose judgment and wisdom is greater than his own.
3) And also, the being evil person believes that ruthlessness cruelty and destructive acts are right and correct.
4) Hence, in the context of law, he believes the health of the community is best preserved by ruthlessness and cruelty, even toward other members of the community. Only in this manner can the community be kept fit and free from influences that might weaken it. Only in this manner can the community as a whole be made strong enough to assimilate all others.
5) Hence, the goal of the community is not the immediate weal and happiness of any member of the community, nor the health of its individual members. If members of the community have to eternally suffer to ensure the health of the community of the whole, or even if the majority of members of the community have to eternally suffer to ensure the strength of the community, then this is an acceptable outcome provided that the community survives and triumphs over all others. The goal of the community is like that of 1984's EngSoc, "a boot stamping on the face for all eternity".
6) No contradiction between these beliefs is obvious, and also we can observe that real people hold these beliefs. For example, we can find examples of people who are willing to sacrifice themselves in an evil cause, with no real hope of direct personal gain, merely because they believe in the worth of the cause. Thus, regardless of whether we personally judge this viewpoint as wise or correct, the term LE is descriptive.
In the past, you've advanced the idea that the term "good" is proscriptive. In the alignment system, it is not. Likewise, you've tended to equate "good" with "serving others interests" and "evil" with "serving your own interest". That description is a rather late development in describing alignment by later writers for TSR/WotC, and is I think in error. If you drop those assumptions, the alignment system is much more coherent.
On that approach, the LE person is not committed to law. Rather, s/he has a belief that order will let him/her impose his/her yoke upon the world. The "L" part of the LE alignment isn't a value to which the LE person is committed. It is a marker that the person believes that social order will secure the person's self-interest (just as, for the LG person, it is a marker that the person believes that social order will secure G).
Only if you define self-interest in very abstract terms. For example, you might argue that a person who sacrificed their own life with no hope of a happy afterlife or any other sort of gain, but merely to advance the goals of society did so because ensuring the continuation of his heirs or cousin or community extended his own self-interest even if he was not to be any longer a part of that society, but I'd find that a very strained explanation for his actions.
I don't actually believe LE persons are actually motivated by self-interest and are merely pretending to advance the notion of order because they believe it is of benefit for them to do so directly. For one thing, that implies that all LE persons are mere hypocrites. For another, that would imply Belkar Bitterleaf has converted to LE, despite the fact that he still acknowledges nothing but his own self-interest and no constraint upon it. I think it's not unreasonable to imagine an evil society with members that are willing to sacrifice all their own interests and even their very being to advance the society as a whole. Your conception seems to argue that its not possible to act except in one's own self-interest. Is it impossible imagine a self-sacrificing evil? The truly LE ruler would I think sacrifice himself for what he saw as the good of his society, provided of course he was convinced that their was a sufficiently strong and ruthless heir by which and under which the society could continue.
Earlier you correctly asserted that LG sees its ideal society as one in which the lower ranks of the hierarchy are content, happy, healthy and uphold the society through their wellbeing and rightful gratitude for the state of affairs. Is it impossible to imagine a society that sees the ideal state as one in which the lower ranks of the hierarchy are oppressed, exploited, and shackled and who obey ought of fear, and uphold the society because they don't have a choice. Is it impossible to imagine a society that sees the exploitation of the masses - to force them to give more than they'd ever choose to give themselves to give them no choice in the matter - as the path to the strongest possible society?