Property is not a positive right. Property ownership is derived from the self-ownership of the person who created the property by combining their own labor with unowned land. The exclusive right to decide how the property is consumed is a negative right; others have no obligation to provide anything to the property owner, only to leave the owner (including their property) alone. Homesteading imposes no cost on others for the simple reason that these others have no claim to the unowned land being homesteaded.
Your error is starting from the assumption that land is owned in common by everyone, rather than unowned. If land were actually owned in common then you would need to obtain permission from every single person on the planet before using any of it. You would starve to death long before you obtained even a small fraction of the necessary consent. And no, the government cannot grant that consent on behalf of others who never deliberately and voluntarily agreed to permit the government to represent them. You would need the consent of each and every individual.
Your error is starting from the assumption that land is owned in common by everyone, rather than unowned. If land were actually owned in common then you would need to obtain permission from every single person on the planet before using any of it. You would starve to death long before you obtained even a small fraction of the necessary consent. And no, the government cannot grant that consent on behalf of others who never deliberately and voluntarily agreed to permit the government to represent them. You would need the consent of each and every individual.