Why are android permission so broad?
As an Android developer, this is an especially frustrating aspect about delivering apps into mass market.

The wording on permission models have caused great grief, as you have pointed out, but it has gotten better.  In fact, prior to Android 5.1.1 (API 22) apps were required to request all of their permissions up-front, at install time.

Now, permissions are generally requested when they are needed - i.e., when a protected API is accessed at runtime - allowing developers in-app time to explain their intentions.

If the developer of an application were respecting the spirit of permissions enough, their applications would, ideally, still function without being granted a particular permissions.

Obviously, if you deny a photo app access to, say, the Pictures directory, the app experience might not be stellar.  However, an app should still continue to function - that is, without crashing - regardless of granted permissions.

Also, as others have pointed out in the comments, developers and publishers are encouraged to provide meaningful insight into why their app needs the permissions it is requesting.

Now, focusing on your main concern:


  . . .why aren't permissions more restrictives/specific ?


They are becoming more restrictive/specific.  For example, Google has introduced Scoped Directory Access.


  Apps such as photo apps usually just need access to specific
  directories in external storage, such as the Pictures directory.
  Existing approaches to accessing external storage aren't designed to
  easily provide targeted directory access for these types of apps. For
  example:
  
  
  Requesting READ_EXTERNAL_STORAGE or WRITE_EXTERNAL_STORAGE in your
  manifest allows access to all public directories on external storage,
  which might be more access than what your app needs.
  


Note the bullet point - relating to the game you mention, the developers could leverage scoped directory access models to fine-tune their permission requests.  That, in tandem with clear and concise permission explanations, would strengthen trust for app's users.

Why haven't they, as well as many other apps out there, already utilized this feature?  Well, the scoped directory APIs become available in Android 7.0 - meaning only devices running Android 7.0+ can actually use it.  It is often simply a matter of prioritization for developers on whether or not to implement different behavior on different devices.  Personally, I believe in doing what's best for the platform, but I digress.  Feel free to reach out to the app's support channels requesting changes or deeper explanations for their permission requests.