911 is normally operated by PSAP, under city or county jurisdiction, not the police department.
BTW, liability exists both ways. The Martin family could sue the 911 operator for not being assertive enough in telling Mr Zimmerman to stop pursuing his "suspect".
Furthermore, 911 lawsuits are fairly common and some states grant full immunity to 911 operators (don't know about Florida, but NJ definitely does) exactly because of that.
All in all, your theory of "wanting to avoid liability" reeks of uneducated guess.