I disagree that PoD is a hybrid PvE/PvP covenant, and consider it primarily a PvP covenant with only a minor PvE aspect. For a few reasons:
1) Invaders block the host's progress; fighting invaders becomes a requirement for the host to progress and gain further ranks. The host is literally unable to go past the fog wall at the end of each zone if even a single invader remains in the area, regardless of whether there are any NPCs alive. If the covenant weren't so abandoned (from people treating it as a PvE covenant instead of a PvP one) this would make PvP a required component for raising the covenant rank.
2) The covenant has unique invasion and pvp mechanics that do not exist anywhere else in the game. Were it exactly the same as a normal invasion, then I might agree that it exists only for lore/flavor purposes. As it is, abyss invasions are so different from any other kind of invasion that I simply cannot treat it as if it were a mere afterthought, and rather that it had to have been an intended part of the design. They even took other PvP covenants into consideration for this one; having ranks in BoB or Sentinels lets you keep either aura when invading in the Chasms (even though blue auras don't normally appear when using red orbs), and wearing Dragon equipment changes your phantom color to look like a dragon spirit instead of an abyss one, a red invader, or a blue invader. Referencing other PvP covenants like that, and carrying over what is normally a cosmetic reward for two of them, goes just a teensy bit beyond the bounds of simple lore, methinks.
3) PoD is not the only covenant with a boss at the end of it; Bell Keepers also has a boss. Granted, Keepers advance their covenant solely through invading, whereas Pilgrims advance their rank solely through area progression. But, considering point #1, this would not make Pilgrims any less of a PvP covenant under normal circumstances. Were Pilgrims of Dark as populated as Bell Keepers, then the host player would need to fight other players every step of the way, and far more simultaneous hostile players per zone than any other location.
4) After defeating the Darklurker, all three abyss gates are permanently opened for free until the next NG cycle. Since the Darklurker is already dead, the only conceivable reasons to go down there are to farm Ascetics or to invade. Invading also happens to give ascetics, and is sometimes an even faster way to farm them on a busy day. In other words, unlike other covenants where invasion is a necessary step for gaining ranks, in PoD invasions are treated as the reward for completing the covenant. This, if anything, makes it more pure; PvP becomes the goal, rather than a means to an end. The only people invading there are people who want to invade there, for the sake of invasions.
Sorry if this is a bit lengthy.