Somehow, it doesn't seem to be possible to plan post for my subscibers (visibility private) as they get published private automatically as soon as I save them.
Is there any way I can keep wordpress from doing so?
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The problem is that changing the post visibility to private invokes a status change, in a way private is not only a visibility but also a publishing status. Or to quote the Wordpress Codex, section »Content Visibility - Private Content«.
Once you change the visibility to private, the post or page status changes to "Privately Published" as shown. Private posts are automatically published (...).
To achieve what you want you probably have to work with the publish_post or even better the publish_future_post hook. Another possibility is to work with Post Status Transitions and the according hooks.