I agree with all the problems that you listed, but I believe the spreading of rumors is a problem that has to be tackled. It is true that there are many more problems to attend to, but why not fix them one by one, starting from the most obvious ones like rumors.
I'm more interested in the idea that "rumors" are a problem that needs to be tackled and the best way to do so is to ban them and only trust what is coming from the government. Isn't this type of system ripe for abuse? Let's say a rumor was that a certain government official embezzled $100M Yuan, but it was true. Perhaps that would be very embarrassing for the central party and thus they issue a statement saying that is not the case. Isn't it the public that is hurt in this situation?
However, you might not realize that there are many other rumours in China that are not true, and has nothing to do with the government per se, such as a disease outbreak or a certain foreign firm exploiting fellow Chinese people. These are the rumours that we need to fight.
I do not expect you to believe me though. After all you would not know any of those without living in China for a few years.