That's a great idea,
but politics, like nature, abhors a vacuum. If you eliminated every trace of government, something else, like businesses, or criminal gangs, would move in and take its place.
That was the central thesis of Anarchism, that no one should be in charge, not governments, businesses, the Church, the Army, no one. The People would spontaneously self-organize into People's Collectives which would run everything from insurance companies (some of these were actually successfully implemented!) to Public Admninistration (the Paris Commune). Not surprisingly, both businesses and governments promptly demonized and outlawed the lot. And Libertarians are just Anarchists with property. It would be like putting Dracula in charge of the Blood Bank.