But again, if no coercion could take place in the system, why does it matter what the government does?
Because the nature of government is that is “an institution that holds the exclusive power to enforce certain rules of social conduct in a given geographical area”, and those rules serve the end that “physical force is to be barred from social relationships, men need an institution charged with the task of protecting their rights under an objective code of rules”, thus “a government is the means of placing the retaliatory use of physical force under objective control—i.e., under objectively defined laws”.
There is a broader philosophical point here, that man’s chosen relationship to an existent is properly determined by the nature of the existent, i.e. man commands nature by correctly and non-contradictorily identifying nature. To be more specific about the nature of government, government is that institution which states law objectively, states laws only for the purpose of protecting individual rights, and enforces those laws by applying retributive force against those who violate the law. Furthermore, no other institution may do this: government has a monopoly on the use of force. Because the use of force is a terrible and dangerous thing, the government’s use of force must be very tightly restricted. It can punish an alleged evildoer only after the person’s actions have been objectively inspected by independent-minded evaluators (a jury, for example) and compared to what is prohibited, by law. Your premise “if no coercion could take place” implicitly recognizes this point about the need to severely limit what government may do. As Rand puts it ("The Nature of Government, p. 128):
It has to hold such a monopoly, since it is the agent of restraining and combating the use of force; and for that very same reason, its actions have to be rigidly defined, delimited and circumscribed; no touch of whim or caprice should be permitted in its performance; it should be an impersonal robot, with the laws as its only motive power. If a society is to be free, its government has to be controlled.
This has an immediate logical consequence about freedom and the “burden of proof” or need to justify an action: the individual need not justify any of his actions, and the government must justify all of its actions. Thus
Under a proper social system, a private individual is legally free to take any action he pleases (so long as he does not violate the rights of others), while a government official is bound by law in his every official act. A private individual may do anything except that which is legally forbidden; a government official may do nothing except that which is legally permitted.
Now in fact it is not and cannot be the case that government is prohibited from using force. On the contrary, it is in the very nature of government that it uses force, in a highly controlled manner. Therefore your suggestion that it might be okay for the government to act arbitrarily as long as it does not use force denies the fundamental fact that government is the agency that is supposed to use force, for a very restricted purpose. This is the contradiction that you need to eliminate. The way to eliminate the contradiction is to identify the premise that you hold that allows the contradiction. I don’t know exactly what you have in mind, but it seems to me that you’re thinking “Government can do anything it wants, as long as it does not use force”. The simplest resolution of the contradiction, I think, is to reject that assumption, and replace it with the premises held by Objectivism. Those premises are that government may only do that which is permitted by law, and that proper law exclusively serves to protect the rights of individuals. It follows that establishing government as a charitable agency is not proper law, since acting as a charitable agency does not protect the rights of individuals.
Note, specifically, that “maximizing freedom” is not the purpose of proper government, it is a consequence of having proper government. If “maximizing freedom” were the purpose of government, then government would simply not exist and you would be free to rape, murder and pillage.