Hired,

My way is different from yours but I have never had problems in China.
Surely, one needs a three-month tourist-visa to have enough time to look for the right school. Three years ago, I checked about three or four schools until I took the final decision.
Foreign affairs officers will not always agree with foreigners because otherwise, they are losing face. ( your opinion ).
In Inner Mongolia, four members from the PSB visited my flat and agreed that the solar heating system there was not enough for the cold and long winters. After they left, nothing changed.
In Jiangxi, all foreigners asked for an air-conditioner in the living room. ( not only in the bedroom ). No positive answer from the FAO.
Actually, it depends on the sum of money involved.
I have been teaching at the same college in Suzhou for two years. I have an air-conditioner in each room, cable TV ( CCTV 9), free internet, utilities paid by the school, return ticket and a 12-month visa and contract. ( full paid winter and summer vacation ). Salaries are always paid on time and the foreign affairs officers organize free trips for foreigners every term. ( very nice and polite guys )
In conclusion, looking for the right school takes time and can`t be done properly on a Z-visa. If you are a runner with a Z-visa or residence permit in your passport, the FAO will complain to police and your name and details will be placed on a special website.
Later on, you can`t get a residence permit anymore ( this is a new measure in China ).

Best wishes,
CAT