The article you linked mentioned how that approval rating (for the central government - not the local ones) came to be for rural people: Censorship and propaganda combined with an attitude towards government similar to what you often see with religious people. If something good happens, the big guy far away did it. If something bad happens, it's due to the corruption of men (in this case the corrupt local officials).

Edit: From the article:

“I think citizens often hear that the central government has introduced a raft of new policies, then get frustrated when they don’t always see the results of such policy proclamations, but they think it must be because of malfeasance or foot-dragging by the local government,” said Saich.

Compared to the relatively high satisfaction rates with Beijing, respondents held considerably less favorable views toward local government. At the township level, the lowest level of government surveyed, only 11.3 percent of respondents reported that they were “very satisfied.”

[..] This dichotomy is highlighted by a 2017 Gallup poll, where 70 percent of U.S. respondents had a “great” or “fair” amount of trust in local government.