This is literally the definition of government used in e.g. social science: https://en.m.wikipedia.org/wiki/Monopoly_on_violence

> private citizens have the right to use force under various circumstances.

Yup, as defined by the government. You can defend yourself in certain circumstances, but you are always subservient to the military and police. They are the ultimate power in the state. If there was a force stronger than the military or police, that power would then be the state.

> The term state refers to a form of polity that is typically characterised as a centralized organisation. There is no single, undisputed definition of what constitutes a state.[1][2] A widely-used definition is a state being a polity that, within a given territory, maintains a monopoly on the use of force, but many other widely used definitions exist.[3][4]

https://en.wikipedia.org/wiki/State_(polity)

It is probably the most common definition. As per this definition, the citizens can use force because the state grants them that right.

Is the state's army economically funded by a contract established voluntarily?

Can you start a private army to compete (not meaning fight it, but to provide the same services) with the state's army ?

If you have to pay for it (you actually pay for the army to force you to pay for it...), and you can't compete with it for the services it provides, it's not only a monopoly, it's also a tyranny.

not really. you are just too poor. if you had the means to create the right connections, you could be paid by the army, like blackwater.