Apple blocks private APIs because they don't want to maintain their compatibility across OS releases and don't want third party apps to break when those APIs change.
Edit: I'm starting to suspect that people don't know what "private API" means, so I want to lay it out real quick. Apple ships a bunch of dynamic libraries with the OS that apps can link against and call into. Those libraries contain functions, global variables, classes, methods, etc. Some of those are published and documented and are intended for third parties to use. Some are unpublished, undocumented, and intended only for internal use.
The difference is documentation and support. The machine doesn't know or care what's public and what's private. There's no security boundary between the two. Private APIs do nothing that a third-party developer couldn't do in their own code, if they knew how to write it. The only way Apple can check for private API usage is to have a big list of all the private APIs in their libraries and scan the app looking for calls to them. This is fundamentally impossible to do with certainty, because there's an unlimited number of ways to obfuscate such calls.
Functionality that needs to be restricted due to privacy or security concerns has to be implemented in a completely separate process with requests from apps being made over some IPC mechanism. This is the only way to reliably gate access.
Apple's prohibition against using private APIs is like an "employees only" sign on an unlocked door in a store. It serves a purpose, but that purpose is to help keep well-meaning but clueless customers away from an area where they might get confused, or lost, or hurt. It won't do anything for your store's security.
"Private APIs do nothing that a third-party developer couldn't do in their own code, if they knew how to write it."
There are a million things under the sun that private APIs have access to that wouldn't be possible with the use of public APIs alone, good developer or not. Prime example: "UIGetScreenImage()". That function allows you to take a screenshot of the device's entire screen, your app, someone else's app, the home screen of iOS. That's a pretty big security hole, is it not?
There are countless examples just like that one hidden inside the private API bubble. Things the OS needs to function, (although the OS may not need that particular example anymore) but could cause massive security issues.