Zero tolerance policies were demanded by parents who wanted to address the symptom (bullying), not the problem (their child).
I doubt it -- every single "zero tolerance" policy I've seen on any subject (bullying included, but hardly uniquely) has been proposed by administrators largely as a way of minimizing their own responsibility for applying discretion appropriately to individual circumstances (and, particularly, of making it harder for them to be challenged for inappropriate punishments meted out for reasons, such as racial/ethnic discrimination, other than the officially stated one.) Sure, they are often motivated by parents demanding that something be done, and "zero tolerance" is often an impressive-sounding way of saying you are doing something, but the choice of that something is all about having a shield to hide behind, not actually being effective (or doing what people want, for that matter.)
And, in practice, the actual enforcement is often as arbitrary and poorly-exercising discretion as prior to zero-tolerance policies, but its harder for those arbitrarily punished to challenge the actions because instead of showing that the punishment was excessive in their particular case (which, under zero tolerance policies, it usually is not on its face), they need to actually be able to show that the supposedly-universal policy is inconsistently applied, which is much harder to do (because its hard to even get access to the evidence necessary to show this.)