Okay, so how about this:
Government, as in social regulation, like manners, rules both implicit and explicit emerge in any count of humans greater than one. Thus it can be conferred that government is an endemic quality of humans. This can be, and has granted the conception of property and property rights in various forms. It needs no funding, nor formal institution, only public assent. Markets form naturally without the state: barter, debt, and spontaneous exchange (think of ideas, techniques) occur independently and ultimately have very little to do with property rights.
Then there's the "state" (frankly a disingenuous term) which holds a monopoly on violence. With that monopoly they can enforce property concepts of a wide variety. The state maintains power with assent from a power plurality. Of course with that very same power the state government can control a much wider gamut of human interaction. For example they can mandate education at the threat of either jailing a parent for charges of neglect, or assuming the child as the liability of the state government. They can also fund and direct the curriculum taught in schools.
Graeber, in his book Debt speaks on the emergence of money (which itself can emerge independently) this is typified by accepting only one medium to reconcile taxes, specie, which itself is state government issued. The logical consequence of this is that the specie would be necessary to everyone within the authority of the state, and so it assumes the nature of universality. Of course with the position of authority the state government assumes they can mint and debase currency. He also argues that this system was put into action to simplify military campaigns. The predominating mode of exchange previously being debt between trusted parties, otherwise neighbors. Remuneration from a strange soldier who could abscond or die is a risky proposition.
But we can have one without the other. The state itself is predicated on a virulent, and thus, circular fiction, and we've actually got very little in the way of a true impression of precisely how it emerged. One common interpretation is mutual defense. In the fashion of a circular argument it self-amplifies. A neighbor becomes a state, upregulates a military, and so the reasonable course of action is to tit-for-tat and form a state government, upregulating a military. And so forth. Likewise the same can be said of capitalism and acquisitive behavior in general, which are favored by the state and business which also form another amplifying system.
>Governments used to be the only way for large groups of people to work toward a common goal, before private companies were allowed to form.
All the above considered, yes, but only the loosest sense of government, which can be dictated by family, culture, religion, or even business, which itself can operate without a formalized state government.