I can completely understand that Charles would seek reconciliation and be willing to forgive Harry for all he has done in the last two years. This is what any parent in the world would do for their children; still love them no matter what.
But - Charles isn't just a father and his isn't an ordinary family. He is the King of the United Kingdom; William is the heir.
Harry is continuously attacking them and describing them as careless, selfish, full of anger, violent, cold - not to mention the racism accusations. The consequence of this is that he's saying the actual King and the future one are unfit to reign, especially William, who seems to be his favored target.
Harry is not only attacking his brother, he's attacking and slandering the future King.
Why should the King of the UK accept that? Why should he accept that someone, be even his own son, would try so desperately to instill the doubt that William could not be a proper, decent King (and even, for heaven's sake, an unfit parent)?
Harry's saying that he has a lot more material about his family that he could disclose is not just a threat, it's a clear blackmail. One should never give in to blackmailers, ever.
Charles will forgive Harry (not so sure William will); but in my humble opinion, he shouldn't allow him to ever take part again to any formal, public ceremony. All ties as to that matter should be cut.