9.3 hasn't been released. Apple could simply take the health angle themselves.
And why would that angle matter at all? Just because a company cares about health and wellness doesn't mean they should be allowed special privileges. The company I just made up cares about your health and will turn your phone off at night to keep from waking you. I demand Apple give us root access to do so! That claim is just as valid as the one f.lux is making.
Apple can do whatever it wants. I don't think Apple ever promised anyone the AppStore would be fair, and Apple has routinely denied apps for features implemented in the OS later. They can do this, but I think it's wrongheaded in this case, and a bit harmful to Apple in the long term.
First, f.lux isn't asking for root access. All Apple has to do is open up one interface command that allows apps to adjust hue, and to run in the background. Just like Spotify playing music in the background all day long doesn't require root access, just like Facebook pulling updates in the background doesn't require root access, just like Flickr auto-uploading photos soon after I take them automatically doesn't require root access. I'm not sure where you got that they want some level of access that would let anyone run wild with malicious intent. Furthermore, hue adjustment can be added to the list of security privileges in iOS. Just like users have to approve apps access to location or photos or address book or whatever, they would approve access to "night shift".
Second, f.lux is clearly interested in developing this feature to a more advanced and complicated level than simple sun-rise to sun-set schedule, with a constant hue adjustment. They want access to the part of the iOS user-base that is ready and willing to use their app in order to take this idea to the next level. Just spitballing here: Maybe they can use the front-facing camera and light-sensor to detect the kind of lighting in your room and automatically adjust the hue dynamically? Maybe they can automatically determine your sleep schedule from your usage pattern and set the hue-adjustment schedule that way? Maybe they have much better ideas? Either way, I don't see how it harms users nor Apple to let them do this.
Third, even if Apple doesn't let them make a proper app, why not let them make a ResearchKit app or something to let them experiment on willing users, just like Apple touts in their ads? Worst-case scenario, it's a waste of time. Best-case scenario is f.lux discovers something good. They can't brag about how much potential there is for scientific and medical research there is on iOS, and then in the same breath deny access to a legitimate and popular company doing research in a medical area for no good reason.
Finally, Apple has this bad habbit of cutting out apps while implementing the same feature. They are totally within their right to do so, but doing so always erodes developer confidence just a bit, and doing so always spends a little bit of Apple's hard-earned good will. I don't think it's worth it.