No first hand experience, but from what I can tell, children of legitimate Z visa holders can send their kids to local schools for a reasonable fee. See shanghaiexpat.

It is also legal for expats to home school their kids (not legal for the locals).

There is a strong sense of entitlement present in American culture which leads to unrealistic expectations when living abroad. This manifests as the so-called 'ugly American'. China is a place where 'zero-tolerance' is literally a foreign concept, and anything is possible with persistence (and good connections). Which also implies the downside -- nothing is permanent or guaranteed, either.

You'll have to pay some bribes but its possible. Generally, expats with kids have packages that include private schooling. Expats without packages usually have to leave when they get kids, though that's probably for the better with this air (the better private schools have huge bubbles, by the way....)