What has the vaccine rollout been like in Chinense cities? Efficient, I'd guess.
Nope.
The general pattern is that the countries that were quick to shut down the virus are slow to roll out the vaccine (well.... okay, it's OFTEN true that this is the case. Much of the EU and some the Americas appear to have done a pisspoor job of both things).
China has shut down the virus effectively, but the vaccine progress there is slow. The estimates I'm seeing suggest it will take till end 2022 to get the whole population vaxed. Admittedly, it's a big country, but then they've also got a big bureocracy to support a vax program.
It seems like public trust in the Chinese COVID vaccines is low, and the Chinese state has been associated with numerous dodgy vaccine scandals over the years.
State authoritarianism goes both ways---it can force through the implementation of policies as we have seen... but it also creates a lower level of trust, and means that the higher levels of government and of state organizations often fail to get efficient feedback on things that they are trying to do, meaning that if you have a problem brewing, the government has a harder time turning the ship around until the problems have become serious. Meanwhile, the fact that the state has to be taken seriously by everyone and will not brook any kind of defiance or questioning from the public means that attempting policy changes can result in loss of face.
Take the one-child policy as an example. It should have been obvious for years that this was going to cause demographic issues, but it took decades for China to finally shift policies; it's now a bit late, because the one-child family concept has burned itself into the Chinese mind as the "ideal family," and so the fertility rate has continued to fall even as the restrictions are being progressively removed.
My point here is not to say that the West has done a great job either--merely that all styles of government have upsides and downsides.
Also, a lot of Chinese are adopting a wait-and-see policy on the grounds that they are not sure about vaccination safety and also worry that immunity may not be lasting, so given the lack of urgency many feel that they'd rather have a last-minute vax when the borders finally open.