I think people are saying "story" when the real problem is "focus".
If you think of other, long-time, popular rides at WDW you have scenes and in those scenes you have something to focus on. With PotC, the story is basically: Some pirate 'stuff' is going down. There's not a huge story there, but there are individual scenes each with their own focus that makes the ride a classic. I can speak of the pirate auctioning off the red head and everyone thinks, "We wants the red head! WE WANTS THE RED HEAD!" There's a connection there.
You have the same deal with BTMRR, Splash, Living With the Land, etc. They all have scenes and all have a focus.
I've just watched the Na'vi River Journey on YouTube so I haven't experienced it directly, but it feels like it's little more than a trip through an alien swamp and, oh, there's a shaman. Imagine the Jungle Cruise with no scenes - just a lot of greenery. Oh, it may be really pretty. It could make you feel like you were in the jungle, but you'd get off thinking, "Ok. I just went on a pointless ride through the jungle..." It's the scenes and focus that make that ride.
Even when you get to the shaman on NRJ, while she's technically neat and quite an achievement in AA technology (I'm not in Pandora to evaluate AA technology) I'm looking at her with all of her movements and song I don't understand and I have no idea what she's doing or why she's doing it. There's nothing to tell me that she's making peace with the Pandora gods or singing an ancient mating song. She's just an advanced, well done AA doing "something". I think that misses the mark. She's the one thing on that ride that's a focus and, as a guest, you have no idea what the point of it is. I'm probably supposed to connect with her, somehow, but she's just a rando in the jungle. Even the head-hunter on Jungle Cruise tells me something and he's not AA at all - just stiff. I look at him and think, "Oh, watch out... The head-hunter is trying to sell me some heads!" The shaman? No idea.
While I was gathering my thoughts on this, after watching the video, it also dawned on me that 7DMT has a very similar problem. There are two scenes and a "meh" roller coaster. The "meh" roller coaster would be fine if you were traveling through multiple scenes. Instead, you have two, really well done, scenes on the lift hill and at the brake station. Those parts of the ride are awesome. On a roller coaster ride, the best part of a ride is the lift hill and brake station. The rest of that ride is just a "meh" coaster not even traveling through a mine but over and under rocks/small hills. The lack of scenes (and it's overall length but I really think it's primarily the lack of scenes) kind of kills it.