Mulling it over, I'm not entirely sure I agree with the view that Pirates doesn't have story. It seems story is defined as a narrated sequence of interconnected events often comprising one or more of: themes, motives & plot lines; narrative structure (beginning-middle-end, exposition-development-climax-resolution-denouement), characters (protagonist, antagonist, supporting characters), and narration or message (the moral of the story).
Main Street, U.S.A. certainly isn't a story because it has no sequence of moments: it doesn't matter whether you go to the station first, the bakery first, the opera house first - there is no building narration, there is no beginning or end.
But let's look at Pirates. Pirates' scenes break down into:
1. A 19th century Lousiana bayou.
2. Abandoned (cursed) pirate caves.
3. A pirate ship bombarding a town.
4. Pirates looting then setting fire to the town.
5. Pirates inside a burning fort (prison/powder room).
6. A 19th century Lousiana bayou.
Whether or not this is a story seems to rely on whether there is a causal relationship between these elements, and I'd say there was. 1 and 6 need to bookend the experience otherwise it would be about travelling to Lousiana, so they're locked in place, while 3, 4 and 5 seem to chronologically follow on from one to the other. (Yes these scenes could be inverted so that we started in a burning powder room, then found out this was because the town is burning, then found out it's burning because the town is being pillaged, then found out its being pillaged because a ship has attacked it, but this is simply an alternate plotting of the same story). Only the pirate caves seem malleable in where they're located, and tellingly this is the only section that HAS been moved around, at Disneyland Paris's version of the ride.
Now, does this story structure support a building narrative structure of beginning, middle and end? I'd say yes: Pirates very skillfully weaves through an introduction (as above: 1), first act (2), inciting incident (3), second act (4), third act finale (5) and denouement (6), building layer upon layer, teasing at things to come. If it had no story, surely it wouldn't matter what order we viewed the scenes in, but Pirates is very carefully built so that the powder room tops the burning town in tension and thrill, the burning town tops the looting of the town in tension and spectacle, and so on back through the attraction.
Within this framework is of the 'business' of what is going on - the skits, the gags, the stunts, and the characters that populate them, which are essentially random - each one is individual of itself, and that was of course the way Marc designed them. Nevertheless, just because it's a lot of disconnected moments doesn't mean it doesn't have a story. I'd actually say Pirates was very similar to Alice's Adventures in Wonderland: that book is essentially a number of disconnected events/moments, but that doesn't mean it's not a story. Pirates is the same, only whereas Carroll's book uses Alice to guide us through, in Pirates we are the central character (directed by the designer using the ride track).
It could be said that Pirates is not a character based story, but I think that misses out that YOU the guest are that weaving character. Even if this is rejected, I still think it's an event based story - despite the fact that guests do find it so hard to articulate the causal-relationships of what happens.
As for the final element of story, I think ironically Eddie (seen as you profess to there not being a story!), your own decision of including the moral at the end of the ride actually lends more credence to the attraction having a story, fulfilling the 'narration or message (the moral of the story)'. By doing so, the attraction was repositioned from 'that's just how it is' happenstance to being a authored text that the creators feel the need to justify: you added purpose to why it has been told, in doing so admitting that it has been told.
There may not be much to it, but I think it's not that Pirates doesn't have a story, it's just that it's a different STYLE of story. Remember how much difficulty Walt had in turning Alice's Adventures in Wonderland into a film because of it's story structure? It seems that story structure is exactly the type that works so well in theme park rides.