By letting the clock run out, they are probably trying to get a one year visa for the price of a less than one year visa.
The problem in these situations it is impossible for an external observer to know if they are cleverly gaming the system, incompetent, pathological liars or all of the above.
I call this the Laowai Uncertainty Principle. You would ordinarily be correct to assume they are doing everything in their powers to have your documents so they can earn income from your teaching. And this is usually, or at least often, the case.
However, more often than we would like to see, an entirely arbitrary and tangential intervention occurs for reasons that you will never fully understand but in reality is due to incompetence, rivalries, mis-communications, shift in enforcement focus, change of loyalties, etc. ad infinitum.
My suggestion is to print hard copies of your email exchanges with the school and if the clock does run out, present them and yourself to the local PSB.
Best of luck.