The movie doesn’t address this directly. I suspect the manga it’s based on does—I’ve heard it is a vastly more detailed world and plot—but I haven’t read it.
Really, a lot of the reason could easily be meta and narrative—Alita participates in motorball because the author/director want her to because it’s and sets up some other narratives later on. Until a sequel is released, though, we won’t be able to really see what those are, or how much the movie explains things—after all, the end of the first movie was rather like an epilogue and doesn’t explain much of anything.
That said, I think there are some plausible justifications.
There’s no particular reason to think that the whirling blades we see Alita clear are the only defenses; in the flashback, it certainly seems like the soldiers are expecting this (one refers to it by a name, indicating they were aware of it and had a name for it) and worse. Alita cannot ascend with complete confidence.
Nova doesn’t consider Alita to be a very real threat. He’s toying with her, looking to get something useful out of her, but the idea that she might actually make good on her threats? Not much of a consideration. Alita is banking on this to keep Nova honest with respect to the motorball championship.
In addition to wanting to play ball (har.) with Alita for the sake of getting what he wants from her, and not having much concern for any threat she poses, Nova has appearances to maintain. Doubtlessly he could deny her entry despite her motorball victory, and make up some explanation for it, but it would damage his reputation and could risk an upset the delicate balance that keeps everyone under his thumb. He likely considers that a greater risk than anything Alita can personally do if allowed into Zalem. Alita can probably surmise this, as well; she has some sources with respect to how things work in Zalem, and she knows how fervently motorball is treated in Iron City.
The motorball championship is very public. Everyone in Iron City and Zalem knows how this is supposed to go. Nova has to choose between two risks: the risk of allowing Alita into Zalem, which he considers minuscule if he considers it at all; or the risk of disrupting the system that’s worked for him for so long, which is also something he probably has confidence he could handle, but it would take more careful handling and could potentially put him in a worse position. Thus, by his likely estimation, the expected result of denying her is worse than the expected result of allowing her. And Alita and Ido can make the same analysis that I have and come to the same conclusions, and decide that motorball is the most likely way to get into Zalem and in a position to do something about Nova.