NO. In most cases it is NOT AT ALL LEGAL to work on ANYTHING, AT ANY TIME, ANYWHERE in China other than a Z visa.

But...there are a few RARE exceptions to this.

1) Somewhere down south in China you can legally work on an L visa and after 180 days it is upgraded to a Z visa locally. I am talking about (spelling uncertain), Gway-Lee or Gwee Joe? I have seen it in writing before and I know of a few people that had done that in the last 1-year or so, but I cannot remember the facts right now.

Z visas in some parts of China can still be gotten without leaving China's mainland.

2) In some rare situations depending on the structure of the "school" license and depending on how their license is worded and where they are doing business, you CAN and you MAY legally work on an F visa. I cannot not quote a legal source here but I know that it is true on some highly rare and unique situations. Again, I cannot remember facts, but it was something like a business consulting service legal loophole whereby people could work on an F but again, I cannot recall where. I do know it was valid, and it may still be. Some places, during the transition phase of waiting for their right to hire the white monkey licenses, are given a wide birth, even if not officially and the local police will accept or tolerate an F visa pending the Z visa hiring permission from the PSB.

None of these are China-wide policies however.
Some provinces have near total autonomy to do as they please.

In Inner-Mongolia as an example you need a police clearance from your own country to get a Z visa. At least that was the way is used to be.

You also should know that in some situations, the local relationships between a school and the police can make something that is illegal, legal with a little red envelope and everyone is happy from the fire chief to the immigration man. That is also more often the case than not.

Best bet is to get a Z visa in all cases and the FEC and then you are covered 100%

Many yeas ago I was on an F visa, a 2-year F mind you, multiple entry, and I had an FEC at the same time. Stranger things have happened in China.