On the other hand though, I see a lot of problems with a privately run police force. For example if the chief of police was making a profit from fighting crime, why would he not expand his business by creating more crime to fight?

Only if you pay him by criminal caught, as opposed to making him part of an insurance company that is responsible for reimbursing people victimized by crime.

The nature of the service and how fundamental it is to society.

Food is fundamental to society, should all food production be government controlled?

If the only drinking straw company decided it was going to make gold straws, poor people wouldn't get any- but that wouldn't be such a big deal.

If the only drinking straw company decided it was going to make gold straws, another company would get into the straw making business and start making affordable straws.