They can? Even local chinese kids without 户口 can't go to local schools, to hear them tell it. That was one of the primary reasons students came to the school where I worked in 2010-11.
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.