As noted, Zack's time-stopping can affect the world of the show in real ways. We cannot read it as merely a narrative device. We have to accept that, in the world of the show, Zack really can stop time and move people around, just like we have to accept that Wonder Woman's lasso compels people to tell the truth. And we can't really call this a "natural ability," right? So it follows very naturally from the premises of the question—which are simply the facts of the show!—that the answer is yes, Zack possesses a supernatural ability.
But this doesn't really felicitously describe the situation. Saved By The Bell is neither a realistic show (like, say, Gilmore Girls) nor a grounded, internally consistent show with supernatural elements (like Buffy the Vampire Slayer). With its vaudevillian dialog, over-the-top plots, and pervasive goofiness, the world of Saved By The Bell is not very grounded or internally consistent. Besides Zack's time-stopping abilities, the best example is probably how Screech, while in high school, built a self-aware robot that speaks perfect English and nobody especially cares.
Another thing that Saved by the Bell analysts have to keep in mind is that, although it's certainly still appreciated by fans of various age groups, it seems clear that the show was written as a kind of wish-fulfillment fantasy about teenagers, for a target audience of pre-teen children. If we approach the show from that perspective, it's easier to recognize that Saved By The Bell operates under cartoon logic—and storylines like "Screech's Spaghetti Sauce" become much easier to accept when we view them through this lens.
So, in the same way, we might ask whether Wile E. Coyote has a supernatural ability to survive being blown up repeatedly, or whether Mickey Mouse has supernaturally autonomous ears. But these questions don't quite meet the characters on their own terms. Looking at stories in terms of "supernatural abilities" assumes that the stories in question feel beholden to a sense of realism, which Saved By The Bell does not.