The paladin of tyranny is from Unearthed Arcana, a book of variants and semi-official houserule suggestions. Despite being an official publication, it doesn’t have the same level of polish and editing that you might otherwise expect. Several things in the book were really closer to just ideas and inspiration than they were to fully fleshed-out rules. Quite a number of things in there weren’t even written specifically for Unearthed Arcana, but rather were just free things that people had written and Wizards of the Coast collected.
So this omission from the paladin of tyranny’s aura of despair feature should be treated with heavy skepticism. In a rules-as-written setting (read: typically a theoretical optimization exercise or similar, rather than actual gameplay), you might be able to argue it, but for practical purposes it’s most reasonable to assume that the paladin of tyranny’s aura of despair was meant as a replacement for aura of courage in its entirety, and it’s only “otherwise [identical] to the paladin’s aura of courage feature” in the context of external effects, like feats or whatever.
In many ways, the paladin of tyranny is a base class version of the blackguard, and that class also gets an aura of despair that also reduces nearby enemies’ saving throws. There is no mention of aura of courage in the feature, however. It is most likely that the authors of Unearthed Arcana meant for it to function the same way.
And on a personal note, no, I would never allow a paladin of tyranny to effectively have their cake and eat it too by having aura of despair also have all the effects of the aura of courage. The aura of despair is already a phenomenal class feature, quite a bit more powerful than the aura of courage, and it really doesn’t need the boost.