This is oftentimes the biggest stumbling block I have seen for many players new to my table-- the idea that having "levels" does not make your character "better" than everyone else. And this is exactly the sort of idea where my gentle prodding to think of the game we are playing as a movie can get those players to stop thinking that way. Of "game first" for lack of a better term.I also never played the everyone is 0 level approach so town guards were not trivial adversaries until the group was above 7th.
Newer players might learn or discover early in a campaign that "civilians" have the game stats of Commoners and the like. These 1st and 2nd level PCs that have a run-in with the town guard are fairly evenly matched. So far so good. But then however many number of months later, these PCs are now 5th, 6th level... and they think town guards are still merely equivalent to 1st and 2nd level PCs. And thus when they do something illegal in view of a guardsman, they think "That doesn't matter. They are not a threat. I'm 5th level, he's got Commoner or CR 1/4 guard stats, my PC can do whatever they want and the guard can't stop me. I'll stomp him into the ground if necessary." And that's when I ask them "If this was a movie or a novel, would there really be such a wide gap between the abilities of the protagonist and the police officer trying to arrest him? This idea that the PCs are the only ones who see this huuuuuuuuuuge bump in ability week-after-week, month-after-month just because of "earning XP" and "leveling up" and the like?"
Now granted, reminding them of this can take a little bit of wind out of their sails when they think that as they level up everyone around them is also leveling up (and thus everyone remains equal to each other, and so then what's the point of leveling?)... but that's where the emphasis on "narrative leveling" can come into play and allow them the feelings of heightened status. If the PCs can "get their way" when dealing with the town guard, it should be because they've earned the guard's respect within the story and have proven themselves to the town such that they are granted more leeway with the guards than some other Tom, Dick, or Harry in town-- as opposed to getting their way because "they're 5th level and the guards a CR 1/4 and thus mechanically they can beat the guards up".
It's a difference that does tend to widen the scope of ideas the players come up with, when they now realize that every problem isn't and shouldn't be solved just by carrying the bigger stick. Which kind of goes against the grain of D&D in particular, but does make the results more interesting and less repetitive.