I voted in this poll a while back but wasn't really sure how to articulate my thoughts on the topic, and thinking about it more since then, I've managed to refine my thoughts a little. So, in what'll probably become one of my typical essay answers, I'll try to articulate them here.
First off, I voted for atmosphere and world building, but that answer has changed multiple times over the years I've spent in and out of the Bionicle fandom. During most of G1's run, I only had a couple of sets and didn't think much about what was behind them, so back then I would've automatically answered that the sets were the core of Bionicle. I took a break from '06 to halfway through '08 because the '06 sets just didn't appeal to me much at first glance (I thought the Piraka weird and the Inika universally ugly) and from the end of '09 to late 2014 because there just weren't any sets around. Only since getting back into Bionicle, and with a lot of story-focused pokemon gameplay behind me have I really started to put more emphasis on what's behind the boxes on the shelves rather than what's in them. Through this lens, I'd also conclude that G2 was definitely more targeted at 'old me' with its epic sets than at 'new me' with its lackluster story content.
During late '14 and through most of '15, my answer became story/plot for two main reasons: I read all the way through NickonAquaMagna's Nova Orbis and started working on a story in a similar vein myself. Reading through Nova Orbis and in the development of my own story, my first concern was always the plot, the overarching sequence of events that happened and that I wanted to have happen to the characters in the world. However, perhaps somewhat ironically, reading back over my own story in progress I kept finding places where that plot felt contrived, where circumstances just seemed too convenient for the characters, or where there just didn't seem to be much overall meaning to what was going on. It started feeling like I was just getting stuck over and over and that I'd never get to something I considered even worth throwing up online.
Getting into roleplaying games (D&D specifically) halted work on that story for a while as I focused on the characters that I was now playing instead, and it was while working on the backstory for one of those characters that I started to realize that it made a lot more sense than anything I'd thrown together for Bionicle up until that point. The reason for that, I believe, is that rather than servicing the plot, the characters in that story drove the plot. Rather than trying to craft a grand narrative right from the top, I planned that story around the decisions of the characters, focusing on the "why" rather than on the "what." Given their situation and what they knew at the time, did it make sense for the characters in that scenario to make that decision? If yes, than that became way in which the plot progressed. Me moving back to Bionicle in late '15 stopped work on that story before it was anywhere near finished, but when I started working on my 'Nova Orbis' again, I re-focused my attention to the characters and their decisions and ended up with something much better than the first time around.
Work was then interrupted again by The Folly of the Toa, which started out to me as a joke picture, then turned into a writing exercise to improve on the technical aspects of writing (grammar, formatting, etc...), and finally became the culmination of me focusing on and studying characters to drive a plot through. I constructed the entirety of the plot of Folly of the Toa explicitly around the characters' decisions, trying to justify each one as I went along, and ended up with something that I actually felt proud of. Granted, I should've done a lot more editing on it in retrospect, but over the year that I spent working on it, that story cemented the idea in my mind that the plot should always, always be subservient to the characters. If the characters are poorly developed, they're harder to care about and their decisions and actions through which the plot advances become contrived, even ridiculous. Of course it's true that most Bionicle sets through both of the theme's iterations depicted only characters; it's the nature of constraction to feature characters far more than locations, which are the realm of system. CCBS isn't called the "Character and Creature Building System" for nothing. Between that and my own studying of the theme's characters, I probably would've answered that its characters were the core of Bionicle if this question had been posed to me last year.
What caused me to move over to world building and atmosphere in the end was thinking back, not to what I have personally focused on as of late, but to what got me interested in Bionicle in the first place and what kept me coming back for one reason or another: the world and its atmosphere. Why? I think it'd be easiest to explain if I put forth my favorite artifact from Bionicle for each category. In sets, 2015's Master of Water is my runaway favorite. Yeah, the weapon is ridiculous, but that set's just so good at being posed in ways that convey attitude and I think that its mask is one of the best designs Bionicle ever saw. I've had that set somewhere on my desk for the last two years and I'm still not tired of looking at and periodically re-posing it. However, it's nothing to me beyond eye candy; I don't care for the character it's meant to portray (G2 Gali) and honestly don't think it even matches that personality very well. I like looking at it, but it's not really inspiring anything beyond that.
My favorite plot is '07's storyline, because it is markedly deeper than and different from the plots of most of Bionicle's other years. In '07, the Toa were in over their heads and losing until the very last moment. It was the only year in which I could say that the final victory of the good guys was a pyrric victory, not just another happy ending, and that made it so much more interesting to me. That said, that power largely comes from the sacrifice of a character, so... even in my favorite plotline, the characters dictate and are pretty much the core of the plot.
My favorite character is Makuta Krika. Yes, I've got a thing for tragic characters in desperate situations, and even though he wasn't around for long I felt like Krika as a character added nuance to the typical good vs. evil narrative that I never really felt was there before in Bionicle. The fact that he was coming from the evil side made that a lot more significant; we'd seen heroes cross into the grey area before, but rarely the other way around. Still, while I'm sad that Krika went the way he did (and some of my first Bionicle fanfic ideas were terrible, wish-fulfillment type things where he didn't actually die and ended up doing random things during the Reign of Shadows), an image of Krika is hardly the first thing that comes to mind when I think about Bionicle.
My favorite atmosphere/worldbuilding items are the MNOGs. They did such a good job of establishing the island, the world of Mata Nui. While only the first MNOG really followed major parts of the main plot, both introduced many characters and locations along the way, and by being interactive to the degree that they were, they created a vivid world that even the rest of G1 could never quite match. The MNOGs cemented what Bionicle looked like, and just as importantly to me, what it sounded like. I still get goosebumps when I hear that ominous beach chant, the Onu-Koro theme with its rhythmic clanging of tools in the background, and that flute melody from the Le-Koro band. Now that's atmosphere. I can't say that I get those feelings when I think about Krika and '07's plot or when I look at Gali's master set and crucially, the MNOGs came long before any of them. I don't think any of them would've even existed if the MNOGs hadn't.
My reasoning on that is as follows: if it hadn't been for the world that the MNOGs and later side media established, the plot of early Bionicle would've been forgettable. Heroes fighting beasts to protect mostly helpless villagers from one big bad guy, all dictated by the needs of the sets. The sets have to be technic-based? Alright, we'll make the characters robotic. We want collectible items? Alright, we'll give them a bunch of masks. The masks and the robotic nature of Bionicle's beings only became interesting because of how they compared to the world those beings were situated in: they were living in tribes, on an island that hardly looked their natural home. Turns out it wasn't, but we wouldn't have known the how or why behind that without all the world building that was done in Bionicle. Without that, I think it would've rapidly gone the way of its predecessors, the Slizers and Roboriders, at best a three-year line notable mostly for its interesting habit of representing living creatures through the technic system. But Bionicle became more than that because it was always so much more than just the sets, featuring a plot and characters that actually mattered and that people cared about because the world was already established.
So in the end, for me the conclusion is one that can probably already be distilled from what a lot of other people have mentioned above: the worldbuilding and atmosphere are the core of Bionicle because they underpin anything else. Without them, the characters and plots would be all to easily reduced to tropes that have already been seen elsewhere and will be seen again, something to read and enjoy once, then largely forget about. I don't think the average quality of G1's sets was good enough to carry the line on its own without interest in the characters that they represented or the plot those characters were shepherded through, and though a lot has changed from 2001 to 2015, I think G2 is a pretty good example of how a constraction line cannot survive on solid sets alone. Constraction lines like Bionicle depend on their characters being appealing to have long-term success, For the characters to matter, the plot that their decisions create has to matter, and for that plot to matter, the world it takes place in has to matter.
A final note on whether Bionicle would work as a system theme and whether it could still be, in spirit, the same as it was in G1. For me, it's a mixed bag. There's nothing inherent about system vs. constraction that would keep a system theme from having as good a world and atmosphere as Bionicle did, but when it comes to selling sets, I'm far more interested in constraction sets than in system sets. Constraction sets display nicely; with Bionicle, I bought a set, put it together, found a cool pose for it, and from then on it spent its time sitting on the desk or windowsill where I enjoyed looking at it. I have gotten a variety of system sets over the years, but only a few of them have had any kind of shelf life for me, and they almost all still employed some constraction elements to make them, well, posable (the hulkbuster and mech sets from Power Miners in particular come to mind). If a system-based Bionicle theme were to come out, the only way that I'd really be interested in buying the sets is if the figures in them were customizable and posable, worth displaying in a way that standard minifigures aren't to me. I don't care much about the grand buildings or vehicles, the static objects that are the subject of most system sets. If a big Kini-Nui set came out with all the Turaga, for example, I wouldn't feel a need to buy the set, but maybe I'd try to get the Turaga on bricklink. However, Ta-Koro Guard battlepacks would fill my desk in a week if the figures in them were decently posable and customizable. I'd build a whole army of 'm.
Already, Nexo Knights has shown that a set consisting only of a figure and accessories is possible in system and is possible to do well in system; constraction is not required, but for me to buy a system set, it has to have the qualities that I like about constraction. It has to focus on the characters, not the place. The theme could use something like a new MNOG, a movie, or a TV show to create those places in far greater detail than any number of sets ever could, and I'd be more than happy having a bunch of more posable Bionicle minifigures sitting on my desk.
I should acknowledge as well that this is only my view, a product of my experiences and particular interests. To prolific MOCists, use of constraction might well be a requirement as it allows for far more freedom in creating intricate MOCs of characters than anything minifigure-sized is ever going to do. I should also point out that the content that I've created as a fan of Bionicle has been far more focused on characters and plots than on world-building; I don't paint landscapes, I draw characters and write stories about and through them. But without the atmosphere, without the worldbuilding to underpin it all, I feel that those characters and plots become meaningless, just another forgettable collection of tropes. In fact, if Bionicle hadn't established that atmosphere and world right from the beginning, I doubt that I'd be writing and drawing about the theme at all.