In my opinion, the public sector has grown so much because people want more and more things from the government. In addition, the interests that are helped by each particular government program want to keep those programs alive and other people lack any serious incentive to kill any programs.
As an example of the first factor, people (especially the elderly) were very much in favor of the Medicare drug benefit. They supported Pres. Bush in passing this and that expanded the private sector.
As an example of the second, farmers tend to want to keep farm subsidies. No one else has much of an incentive to kill those subsidies because the benefit that any individual taxpayer got from killing those programs would be miniscule.