Yeah - you've touched on one of the reasons I'm so grumpy about it. A City councillor has a governance role.
They consider and set city policy and review and approve executive actions. There's not really any scope to be anti-woke or anti-anything really unless you squint really hard. The majority of the job is understanding the concerns of residents and balancing that against the advice of experts and the difficulties encountered by the executive.
It's not really a platform from whence to dispense your authoritarian demands.