We're probably talking about slightly different aspects of the same thing.

Emergencies are handled by pulling peope that are off duty

There are different levels of emergencies, right? So there is likely Y number of officers that have to be on duty at all times, and an additional X officers that number officers need to be employed by the department yes to respond to the highest levels of emergency to be able to pull people in when they are off duty.

firefighters (volunteer and professional), and first-responsders from the surrounding area.

For this discussion, I was assuming we're only talking about police response needed. When I say "police needed to respond to emergencies" I'm only talking about the police component, not the other type of first responders. I don't see how its useful to expand the conversation to other first responders when we're only talking about employed police officers. If you have information about how the number of firefighters/EMTs can increase or decrease the number of police officers employed for policing, I'm open to consider that argument or something similar.

Cutting the police budget is notoriously difficult for political reasons.

I agree with this.