Here is my take, once again restated: player driven campaigns are driven by skilled players. Whether those develop into strong stories depends largely on skill of the players to personify characters and give them strong dialogue and motivations, and to make strong character driven choices within the setting.
While I wouldn't deny that player skill will impact the outcome of player-driven play, I don't think it's the only thing that matters. Any game is created by three things: the players, the GM, and the rules. So yes, the players will have an impact, but so will the GM and the rules.
The GM has to support the idea that the game will be player-driven. The rules have to support that kind of play, as well. The more effort that the GM and the rules make for the game to be player-driven, the less it will be solely on the players.
I think it's rather odd to talk about what you can do as a GM to create player driven campaigns and strong player driven stories in those campaigns. Really, what can you do beside let players create their own choices and just respond to that? A GM can't make a player create a good story. A GM can't make a player RP well. A GM can't make a player create interesting choices that will lead to interesting circumstances. We're supposedly talking about respecting a player's agency but we're doing it from the perspective of the GM making all the choice.
The GM can involve the players in determining what the game will be about. The GM can ask questions of the players, and then build on the answers. The GM can refrain from crafting an entire world and pre-existing stories for the characters to explore and instead can help craft a setting that serves the PCs' stories.
I think a better approach would be to focus on what players can do to drive a campaign and create strong stories. The topic really IMO should be, "What can I do as a player to tell better stronger stories?" And I think if the thread took that approach honestly, it would reveal how weird the original post actually is. Like, can you imagine writing an adventure to be purchased by and consumed by potential players? And like how would that actually empower players? What would a player's story look like and how could you communicate it, and if you did wouldn't it cease to be the player's story? And would the player really want to know their own story? How can the player be the secret keeper?
The original post is not weird at all. It's asking a legitimate question. But everything you're saying relies so much on a traditional GM-player paradigm that it seems you can't even imagine that there are legitimate answers to the question.
Think about it this way. My PC's are bounty hunters working for the bounty hunter guild. Like, I could run the game by presenting the players with 10 or 20 choices of what bounties they wanted to go after. But even if I did that, I'd still have to invent the scenario. Currently they have been hired by an Imperial Prefect to investigate the disappearance of colonists on a frontier world. Now do the players want to know at the start why the colonists are disappearing? And remember, "We are bounty hunters" was a player driven tagline. The current scenario is running at like 22,000 words of notes plus maps, and most of that is documenting what happened before the PC's arrived and so therefore, what history exists for them to uncover so that they can make real choices based on their own investigation.
I think relying that much on backstory... that there's 22,000 words of things that have already been decided... is a mistake for player-driven play. That is the GM committing strongly to a lot before play even begins.
I would also ask that if the players have indicated they want to be bounty hunters, why not present them with some bounties to hunt? Why are they investigating missing colonists?
I would think you'd take the player request to be bounty hunters and then combine that with some other player requests. These requests would hopefully be based on their characters in some way. This is where rules come in... some games have players declare traits or instincts or bonds that the GM would then use in play to make the events of play suit the characters.
I don't have a clue what they are going to do, but I do have to document very carefully what is there because the resulting story has to be internally consistent. You got 10 different disappearances and crime scenes and impacted families and communities that need time investment to describe. And, minor spoiler, there is actually an important element of the timing of the crimes that required a lot of work from me to get the outline of the past right. All that world building is necessary if I'm actually going to create a sandbox that also generates the big linked dramatic story that the original poster wants.
All that work is not necessary. That's a choice you made. And that's perfectly fine if that's what you like and if your players are into it. But the OP is asking for other ways to play... and they do exist.
The underlying problem is that it's not that fun if the same person who introduces a problem is responsible for solving it. I mean that's the problem with a railroad - the GM both introduces the problem and sets the solution. But the reverse doesn't work either. You can't have the player both introduce the problem and its solution because then there is no drama.
The players don't have to introduce the specific problem, and certainly don't need to know the solution ahead of time. The players need input on the elements of the game. I'll offer a few examples of games I've played or run recently that were very player-driven.
Blades in the Dark- the players have a lot of say about what will come up in play. This happens in lots of ways. Their selection of playbooks and playbook abilities are cues to the GM to include related elements in play. The group actively creates a Crew with its own playbook, so that will have a huge impact on play. A game featuring Smugglers will be different from one featuring Assassins will be different from one featuring a Cult. Each Crew also has abilities and contacts and friends and enemies that will influence play. The GM will make a note of all of that stuff, and will then use it as material to bring up in play. He's not going to ignore their choice of Smugglers and the fact that they're enemies with the Red Sashes gang and instead have them looking into ghostly possessions of the city's elite. All the decisions the players make when they create their characters and crew should shape the game.
Spire: The City Must Fall- Spire assumes the players are members of a secret Drow revolutionary group devoted to resisting High Elf rule of the city. Beyond that basic premise, there are a lot of ways the players will influence play. Each Character Class has associations with factions within the city that will impact play, and Class abilities that should also play a significant role. The factions linked to the classes are often also connected with a specific district of the city. Each district has a specific feel to it, each lends itself to certain types of stories. When I ran this game, I asked the players what kinds of stories they might like, and then we chose the district based on that, and they then selected classes. When they make characters, they also create Bonds with both other PCs, and also with NPCs. The ways Bonds with PCs work is to establish the connections between the characters, but does so in an interesting way... each player gets to declare a truth about someone else's PC. You get a strong sense of the shared history there, and any GM will know to include this stuff in play. The players also choose Bonds with NPCs. These are relationships that can help the PC, but which also may put the NPC at risk.... it's all rife with potential for drama and excitement.
Stonetop- The players all make characters who live in the eponymous village. Stonetop itself serves as the binding element of the game. Everything the characters do will be in service of the town in some way. Similar to the other games, the players will contribute significantly to the creation of the town and its inhabitants. They'll craft the existing relationships they have with NPCs. Each will also select an Instinct, which is a huge indication of what is important to their character, and which the GM should be considering at all times in play. Part of character creation, which is always done as a group, is to take turns asking a series of questions. Each playbook has its own list of questions. The players answer these questions, which will largely determine the events of play. The GM should pay close attention during this step, and should offer advice or suggestions when needed, but otherwise leave the players to it. These questions are designed to create initial threats or situations for the characters to deal with. The players are actively involved in determining these elements. They're not determining the entirety of things, nor pre-determining a solution... but they are actively involved in choosing these elements.
Each of these games has a default setting, and a book that goes along with it, but none of them require the GM to write 22,000 words of anything. For my Blades game, my prep consisted of setting the opening Score to get the ball rolling, and then offering a couple of options for their second score, and then after that, I would just use what came up in play to further the events and offer new scores. For Spire, aside from printing some stat blocks (which are minimal), my prep consisted of a one page mind-map that showed the significant NPCs and factions of the district of Red Row. That was it... one page of prep. For Stonetop, I made notes during character creation, noting the Instincts of each character and their relationships to NPCs, and then I crafted an initial situation that built upon their choices and would challenge their Instincts.
In each of these games, I could ignore all that player input and come up with a 40 page backstory that will determine what play would be about, and we could likely still play and enjoy the game. Each could (largely) be played in a very traditional manner. But I'd be ignoring the intentions of the games and their processes. This is how the rules and the GM contribute significantly to a game being player-driven.