You pay for the police all you want, via tax, but you can not force me to pay what you want. And you can't send your hired mercenaries (police) to make me pay for your hired mercenaries. Than you would be violating me and your mercenaries too. And I need protection from you and your hired mecenaries

Police can only do what normal people also would have the right to do....because they are normal people, just like you and me. They have no higher power and they can not get it from you me or anybody else either, you can not give what you don't have, and imagining that by doing some serious looking pseudo religious rituals you can give them that higher power is just a belief (as in religion) but doesn't give them higher power either. That someone who represents you can only do what you can do yourself, can also be show in that you can't hire a hitman to kill someone and give him higher power. You can not give them higher power neither can I. They have the power to stop real crimes, those with a victim. Just like me and you.

And the story of the old lady does not change that.
That is a, what if, problem.
Also that you are being protected, by police, I don't see it. It is really extreme rare that they protected someone (up front) And history shows they did not do much protection. They just follow orders just do their job, which consist of enforcing the whims of the politicians. history shows very well how organized protection always turns to organized murdering of the masses of people they should have protected. But very few of the order followers are actually protecting or turning to the resistance and thereby breaking the law. (because they are order followers they follow orders not their moral compass or conscience)
But of course it could be that, in communistic russia and nazi germany the police did maybe, in the meantime, help some old lady, when they where not buzzy deporting and murdering other innocent people.

Also because you and other people think you "need" them doesn't justify them violating me.

Maybe you can not imagine how the free market would bring such a thing. But I'm not gonna be a slave to your lack of imagination. The free market will not bring it either, we ourselves have to do it.

We are the free market, a market where the initiation of violence is not welcome. Because when there is the initiation of violence, there is no free market.

It's not hard for me to imagine hiring a protection agency or becoming part of one myself, without using violence and threatening my neighbour to pay for one, or pay me, with the use of violence.
We people can organize protection without becoming criminals ourselves, by threatening and stealing and violence.
But It's just a hard concept for other people to grasp and I understand that.

English is not my native language so there may be a mistake or two.

Peace.