The fact that Mainland politics is really old-fashioned and based heavily on "saving face" compounds the difficulty. From the protesters point of view though, it is not their problem - A political apparatus that isn't flexible or modern is a fault of China, not Hong Kong. I think it is an excellent test for Beijing on how to deal with an educated, engaged and motivated populace that doesn't see any reason to respect its legitimacy, because it's not going to be the last time they need to deal with it, isn't it?
This "saving face" is not just politics - it's culture. It doesn't account just for the mainland government, also for the HK government, and even (to a lesser extent) the opposition politicians. For individuals and companies not having to lose face is just as important as it is for politicians. For the central government there's even more at stake, as president Xi has been working hard to cement his power in the mainland, and if he gives in to Hong Kong protests, that could give reason to mainlanders to start protests as well - if the government gives in once, it may happen again.
Many HK people will also not question the legitimacy of the Beijing government directly, certainly not as much as the legitimacy of the HK government is questioned. The mainland government I believe is mostly accepted as a fact, something we have to live with and we're not going to change. It's also seen as a government that should stay out of Hong Kong's internal affairs - let them take care of the foreign and defence policies, leave everything else to Hong Kong. Some may respect that government, others not so much, it's not something the Hongkongers can do anything about.
CY already called the protests of being "out of control". The local daily South China Morning Post correctly added that he probably meant that it was out of his control.
How the officials managed to reach Golden Bauhinia Square this morning for their ceremony, I really don't know. Many roads in the vicinity were occupied, so getting their in their oversized limos would have been an issue. The ceremony has been short, the fireworks tonight were cancelled. No matter what, the HK government lost quite a bit of face here.
Anyway, there are indeed not that many people in Hong Kong that care too much about the central or HK government losing face. Not many will be out to deliberately make them lose face which again is against Chinese culture, however in this case Hong Kong's interests come first, helping them not losing face is a distant second.