The Fed and ECB's goal is not to prop up stock market valuations, but to help the economy by encouraging spending and investment through moderate inflation. The banks are tightly bound by inflation targets. Arguably, the ECB has done too little QE since nominal GDP has barely (or hasn't) recovered in most of Europe since 2007, and inflation has been way under target since the crisis.
It seems like China's reaction is more panic-driven, and doesn't have a framework such as inflation targeting to constrain it. In addition, this intervention is just to prop up stock asset prices; encouraging spending and growth in the whole economy is not the primary goal.