While I didn’t get into D&D until 3e in college, my first group’s campaigns were primarily hack and slash games. The DM would set up the encounters, and we’d kill monsters. When I started DMing myself, my first campaign was mostly improvised (because I’m lazy) but more structured. I didn’t really have a narrative in mind, though I suppose in a sense it did evolve expected directions. Reflecting back on it, I suppose it was a misguided attempt to put the PCs in a world that doesn’t exist for their sake while also imposing my ideas on what should happen on it. There were definitely a few “and you get mass teleported to here” or “you need to go there for help” moments. It was fun at the time, but I wouldn’t run like that today.
Regarding the question: “When they find a magic arrow of dragon slaying in Act One, will there be a Dragon in Act 3 to shoot at?” My answer is, “no, not necessarily.” Actually, it’s more fair to say the question doesn’t make sense because my game isn’t structured into acts. It’s exploration-driven with a PC-set goal, but it’s on the players to navigate to it (whether and how they do is essentially what the campaign is about). My role as a referee is to present adversity while making sure the world functions as it should. I like to take that seriously, which is why my homebrew system (the campaign started in 5e but has transitioned through a few systems to what it is today) delegates certain decisions to the mechanics to avoid situations where I fear I would impose an outcome instead or create the perception thereof when that’s not what I want.
This kind of play is obviously not for everyone, but I’m up front with my players about what it is and what it’s about. I think that’s the best way to go about things (being explicit in intent, so those have certain preferences can find campaigns that will match their tastes).