Prove it. How many rounds did it take? How many HP did they lose? What spells did they use? Did they ever fail an encounter? And how tautological is "well, they aren't dead, so they must have succeeded"?
That's reaching pretty bloody hard to claim that encounters were successfully resolved without a single mechanic being used. It's all pure narration. Which is fine. We all do it. But, it's not proof that mechanics influence world building when you actually aren't using any mechanics.
The debate continues...
So look, you're right, and you're wrong.
You are correct that no mechanics are used, or at least very few mechanics. But it's not about the mechanics per se. It's about probabilities.
The number of encounters per adventuring day is designed around a certain set of probabilities about how likely the party is going to survive an encounter. This is based on difficulty, but it's also based on some assumptions about how the players will handle a given encounter, even if we don't know exactly what the designers were thinking. For example, I suspect the designers think that players tend not to run away, etc. Other things that I've seen quoted one place or another are things like they expect a rogue to be able to use their sneak attack every round, that sort of thing.
The mechanics are designed in part to support those probabilities. For example, the death saving throw is set so you have about a 60% chance of survival without any additional help. That 60% probability comes up a lot in 5e.
You don't need to know how many hit points somebody lost in an encounter. If you want to maintain consistency in the world, then you ensure that the probabilities apply the same to the PCs and the NPCs. So if the probabilities mean that 60% of encounters end in success, then they can expect similar results for NPCs.
It's not always pure narration. For many of us it's narration based on the same probabilities set up for the PCs. Which means that the rules influence the world building. And that's really what we're talking about. It doesn't matter if I played through the actual combats, because statistically I know what the outcome will be. And I can create simpler mechanics if I want to to produce those probabilities for the NPCs to help develop the world and what's happening in it.
The details may be pure narration, like when you meet one in the tavern and they're telling you about their last battle against a wyvern, but the statistics, the part that directly relates to the elephant, is not.
Because it's not the mechanics that determine if the PCs survive or not. It's the probabilities behind those mechanics, combined with the actions and decisions of the players and the luck of the dice. I can make the objective decisions (or even mechanical) decisions about how the NPCs would approach combat which would provide that piece of the probability for each NPC. Just like I can look at the play style of the players and develop a pretty good guess as to how long a given PC will survive based on the probabilities at play in my campaign.