The insurance policy quoted is the one AOPA refer it's instructor/examiner members to.
However, I agree the flying club should have it's own policy. But that policy will only cover the public while in the club premises, not while airside or on other parts of the airfield, nor will it cover while in the aircraft. Those liabilities are the domain of the airfield or aircraft insurer.
Strangely, public liability insurance is not mandatory, but employer's liability is. Self employed instructors are considered employees in this case.