Would Romney admit to a health problem if he could get away with not telling anyone? Xi is not just some politburo member, he is China's next president in waiting, and any confirmation of health problems could hurt him politically just like any health scares would hurt the election chances of a candidate running for president in the US.
That is all assuming that Xi had some health scare to start with, which in itself is nothing but unsubstantiated wild guesses by people who have no idea what actually happened, and the only reason this non-story got any traction is because the western media decided to elevated above the ideal gossip that it is by reporting prominently on it.
It seems that sometimes the western media is not content to merely wait for bad things to happen to/in China so they can cherry pick them to report, they get so restless when nothing bad happens for a while that they have to go and try to find something to the point where they are believing baseless speculation.
The Chinese gave the reason for Xi missing his appointments was because of a minor back injury sustained during his daily swim. If Obama hurt his back and had to cancel some appointments, would the White House feel a desperate need to parade his bed-striken form in front of the cameras? No, of course not. If journalists refused to believe that perfectly reasonably explanation, that' their problem.
All the subsequent hype about Chinese spokesmen not having information about Xi that helped fuel this non-story is more a case of misreporting rather than something the Chinese did wrong. When pressed, the Chinese foreign ministry spokesman said he 'had nothing new to add'. How that could have been twisted into 'the Chinese spokesman didn't know' as was widely reported speaks far more about the utter lack of standards and accountability for work in the western media than some PR blunder by China.