Isn't Nano just saying their approach to this is bad, not that anyone has a right to know her personal medical details.
People and particularly the media, have an unhealthy obsession with the Royals (which the Royals largely encourage when it suits).
Whether you think that's right or wrong is kind of a moot point. The obsession exists.
Lets say she is seriously ill. The desire for privacy is understandable and yeah, I think she has a right to it. But the media don't care and many of the public still want to know what's going on. It's maybe not right, but thats how it is.
Playing this game of secrecy and deceit is predictably having the opposite effect to what they presumably intend. It's just fuelled interest and fed the rumour mill.
They don't have to share all details, but (if she is indeed ill) wouldn't something like "she is suffering from an illness and we ask people respect her privacy at this time" do a better job at squashing some of the rumours? The wall of siletnce just makes them look a bit sinister.
Or maybe William is a true hero and taking one for team (ahem) by allowing some of the rumours to continue to deflect attention away from Kate's illness