The Specialness doesn't have to be revealed right away, or even decided on right away (retcons are a thing, after all), but it was something present in the Chosen One before the story begins. Even had there been no prophecy, their Special nature would eventually have emerged and they would have been important in some way, if not on the level of the Chosen One.
(Sometimes the story will set up a fake Chosen One, who has something obviously Special about them from the start. Then the real Chosen One emerges later on, their position as Chosen One having been obscured by their humble origins or whatever. This is not the same as the Mentor Who Dies, because the story at least acts like the false Chosen One is the real deal for a little while, and the viewpoint character is simply trying to survive proximity to the Prophecy.)
This is not to say that the protagonist won't become Special along the way. The trials they undergo are often of the "what doesn't kill you makes you stronger" variety, and their willingness to follow the path may lead them to Specialness. Perhaps it's a power anyone can gain if they go through the same trials and survive, maybe it's pure luck, maybe they get it from someone who was Special but didn't fall for the Chosen One scam.
Milder versions of "no" can involve latent abilities that would never have been revealed or unlocked unless The Chosen One went through with their trials. Dumbo needed that feather and probably never would have flown without it. The protagonist of Kung Fu Hustle had to get beaten almost to death to unlock his inner power. But at this end of the spectrum, it takes a lot more than just telling someone they have a Special power or background, they need to have already accepted the call and risked their life before the latent Specialness could be unlocked. Even a legacy sort of Specialness needs to be reaffirmed through some sort of trial that the protagonist would never have been able to undergo without accepting the quest first. ("Yeah, you say you're the lost Prince, so do those five guys over there. Prove it.")
Of course, sometimes The Chosen One retains all their power and resources even after the Prophecy has been dealt with. Then the story is about whether they can return to a normal life with all that power, seeing if there's something else the power can be used for, wondering why the power didn't go away, and so forth. Thus, while it was the Prophecy that made them Special, they stay Special even after the Prophecy has been dealt with. If there are more stories to come, they need to find something other than The Chosen One narrative to hang their plot on, though.
As noted in the previous section, if the protagonist remains Special, you can certainly come up with other challenges for them. Perhaps they lost the top end of their power in the aftermath of the final fight of the Prophecy, so future challenges don't need to be as prophecy-worthy. Like, the support agency got blown up a lot, but the core members are still there and aren't completely without resources, so they'd like to help the protagonist meet these new challenges. Or the locus of Fate is gone, but one touched by Fate will always be Special to some extent. A Magical Girl who has broken the cycle of Prophecy may still need to face lesser (or even greater) threats than the eternal foe she'd been reincarnated to oppose over and over. A space hero who saves the galaxy still has a lot of space criminals and leftover warlords to deal with. The farmboy who becomes King now has a realm to rule and some problems can't be solved with a magic sword. And so forth. Put them into some sort of plot where they're challenged, but leave Prophecy behind.
Probably the worst way to go about it, though, is to go back and undo the victory. Say that it wasn't the TRUE Prophecy, but a false "foreshock" of some sort. A false Gathering, for instance, or revealing that somehow, Palpatine lived. Unless the original Prophecy is established as somehow cyclical, it's really hard to follow a Chosen One plot with a new Chosen One plot in the same setting. Saying as part of the sequel that, oh, it's actually an unending cycle...that's cheating.