The paparazzi comes to mind.
Remember when Princess Diane (William & Harry's mother) died in that car crash? She was hounded by the paparazzi, and in an attempt to flee their cameras, her car sped from the aggressive paparazzi, causing her fatal injuries in that crash.
Afterward, the paparazzi were criticized for their conduct, for their part. There were conspiracy theories as well, but her death happened as a result of her being pursued by the paparazzi (people who photograph royalty and sell them).
I think there is a similarity with Catherine's health issues. Of course there is concern from the people who love her, and curiosity since she's famous. But this feeds the hunger of the paparazzi, whose goal is to snap pictures no matter the cost. So, the royals are benefactors and victims of the press.
Williams mother was hounded by this same type of attention, and I'd bet he wouldn't want the attention during this health crisis of his wife, especially when he dealt with this kind of intrusion with his mothers death. So that can factor into why there's so much secrecy. There are times when public figures need privacy - like when there are health concerns or other personal matters.