If you ask me (and you did) the essential problem of an evil paladin is one that has to do with the nature of evil and the nature of a paladin.
In D&D, a paladin is a warrior devoted to the causes of Good and Law. They are not primarily devoted to a particular God but rather to those causes. A paladin is a servant of good and law supported by (some of) the gods.
Now, leaving aside the sticky problem of identifying the ideal of law and differentiating between it and good, it is clear that the paladin believes in and has a cause that is other than himself. A paladin can be forced to choose between doing what he believes in and doing what would be most advantageous for himself. Because good is good, such motivation is possible.
That is not true of evil.
Unless you have a strange moral framework in your D&D world such that good and evil are merely labels for two opposing teams, evil is essentially different from good.
The PH identifies evil with selfishness. Many philosophical traditions would agree with that. Others might identify it with pride. Either way, however, evil is generally self-seeking.
The "champion" of evil would not be a mirror image of the paladin engaged in a disinterested pursuit of abstract evil. Even if there were such a thing as abstract evil, it's not something people would pursue. The truly evil villian doesn't torture people because he believes that torturing people is right or obligatory; he does it because he likes torturing people (or maybe because it's convenient and intimidates his enemies)--he doesn't care about obligations.
The "champion" of evil wouldn't actually champion evil at all, he'd champion himself. It's not comprehensible for the epitome of evil to be forced to choose between his "principles" and doing what's most advantageous for himself; his principles dictate that he do what is most advantageous for himself. (They might also dictate a certain definition of advantage to include an autonomy that means surrendering and really joining the forces of good is never to his advantage however). The character might be an embodiment of evil but he wouldn't be a champion of it. He's a mass murderer, serial killer, or evil necromancer but he does what he does because he wants to and expect benefit from it not because it's "evil."
Because of this essential difference, the abilities of an "anti-paladin" would not suit the embodiments of evil particularly well. The paladin can heal because he wants to heal them. Why should the villain inflict with his touch? He can do that just as well with his greatsword. The paladin smites evil because he lives to stop evil and thereby protect people. Why should the villain smite good? He is just as happy destroying his rival for the title of BBEG as he is destroying a champion of good. The class to represent the embodiment of evil ought to have different abilities than a mere reversal of the paladins' abilities.
What about the champions of evil gods? What about them? Even they aren't engaged in a disinterested pursuit of "evil." They're engaged in carrying out their gods' plans. Others might consider those plans evil but that isn't why the gods are pursuing them. The gods might be trying to increase their power by spreading death or trying to destroy all of their hated enemy's works, or trying to get at another god by destroying people he cares about or something else. Wouldn't some kind of a plaguebearer prestige class be more appropriate to an embodiment of Nerull or Nurgle's will than an anti-paladin? Isn't the fist of Hextor better for a Hextorian champion? (The same might appropriately be said about the champions of good gods).
What about the LE knight archetype. The one who believes that brutality is necessary to hold his enemies in check? The one who philosophically opposes what is called "good" because he sees it as weak? He's not engaged in a disinterested pursuit of evil either though. He's engaged in a pursuit of power or strength or service to the state/collective by any means advantageous. Again, how are "anti-paladin" abilities appropriate here? Wouldn't a general smite (like the destruction domain) be more appropriate than smiting good? Wouldn't a wholeness of body type ability be more appropriate than a touch inflict? The ethos of the LE knight isn't actually opposite to the paladin's. I would expect a slightly modified Knight Protector or Samurai class to be more appropriate to this archetype than an anti-paladin.
**Edit**
Chaos and Law (leaving aside the problems of defining them in a manner that is distinct from our concept of good and evil and that people would actually want to champion) are potentially ideas that could be pursued disinterestedly. So a champion of Chaos or of Law is conceivable. A champion of evil, OTOH isn't conceivable in the same way (except verbally. . . and a square circle is also verbally conceivable).
**End Edit**