Earlier quoted context omitted.
In this case the Chinese government can kill the liveness of the network (preventing it from producing new blocks), but not the safety of the network (they can't fork the chain). Your question has changed into: "doesn't the system break when a malicious party controls 35% of the stake?" To which the answer is yes. If a malicious party can control 35% of the stake, that is very bad. I should add, though, that if you k…
"If a malicious party can control 35% of the stake, that is very bad." That didn't happen in this case. The Chinese government never took anyone's keys or controlled any stake, which is the point of the example. The malicious parties don't need control. They don't need to take anyone's keys or access their servers. They just need to be able to bring it offline, which is much easier to do. Which raises the question I…
It is true that if 35% of the participating stake is suddenly prevented from contacting the network, no new blocks will be created. The word "participating" is important -- you can have stake that is not participating in the agreement protocol, but that is still spendable.
The only way I can think of to mitigate the scenario you've described would be this: before going offline, you can broadcast a transaction which essentially removes your stake from the pool of participating coins until you're able to come online again. This would ensure the network can proceed without you.