Prevent private post 404

You are not seeing a redirect, to confirm your own observation, so much as you are seeing just what it looks like-- a "File Not Found". Take a look at the query in the source around line 2438 (among others). The status of the post is part of the query itself, so if the status doesn't match then no posts are found and that means 404. As you can see from the code, status is pretty rigidly protected. That is proving difficult to override.

I'd recommend using a custom post type for you "logged in only" posts. That is going to be much easier to control both on the front end and the back end.

According to the Content Visibility page, Private posts are visible only to those with sufficient permission levels. WordPress doesn't expose them to non-logged-in users at all, hence (presumably) the 404.

If you want to display a limited preview to non-members, instead of setting the post to Private, you could try something like this:

add_filter( 'the_content', 'wpse101968_preview' );
function wpse101968_preview( $content ) {
    if( is_single() ) {
        if( ! is_user_logged_in() ) {
            $content = get_the_excerpt();
        }
    }
    return $content;
}

And then your non-logged-in visitors will see the post's excerpt instead of the entire post. You can customize the excerpts, too, so that they see only what you want them to see.

References

Codex:

• is_user_logged_in()
• get_the_excerpt()