Well, Apple didn't treat Flux any differently than other developers who used private APIs or tried to distribute binary blobs to iOS users.

Software running in the background is something Apple tightly controls in iOS (battery life and security being major concerns). They also don't let apps make changes to the OS appearance. That may be the wrong approach, but Apple is at least consistent in it. This was either an OS level feature or a no go from Apple's perspective. As a user I'm happy it's a feature!

Yes, that's the whole point of private APIs - they're not supported, and they can be changed or removed whenever.

It's just access control at the framework level. Apple could embed a copy in each application using the framework, but that's inefficient, so they install it at the system level.

It totally sucks that Apple prohibits third-party apps from offering useful functionality that requires deep integration with the system. So no, it is not exactly "OK."

However, it has nothing to do with rejecting apps because of it, and offering their own version of the functionality. Apple has always reserved this domain for themselves from day one, and f.lux would have had the exact same problems even if Apple hadn't been doing their own version of it.