What you describe is the definition of a natural monopoly. However, the monopoly comes from physical wires, not whatever flows over them. So, the infrastructure should be a public good (most infrastructure, really, not just wires, since it's more efficient and less wasteful to build a single network and keep it in excellent condition than to build multiple networks providing the same service at a lower quality), but whatever flows over it can be provided privately and competitively (also, each ISP would have to build it's own servers/routers, etc.)
What you describe is the definition of a natural monopoly. However, the monopoly comes from physical wires, not whatever flows over them.
Part of my point was that there is no physical monopoly here, even on the wires.
BT (formerly the national telephone provider) has one set of infrastructure.
Cable companies have their own, literally installing a different box in your home with an independent physical connection to their own network.
If you use one of the mobile providers, which is viable for a home connection in the 3-4G era if you're only using a modest amount of data, there is no physical wiring to the home at all.
There are even a few special cases where people have literally built their own infrastructure as a community, only hooking in to some big provider's network higher up the hierarchy, presumably on some sort of customised, one-off deal.
So while we do have local loop unbundling and a shared-last-mile kind of deal, and we do have ISPs competing for that same physical connection, even that is not a monopoly provision.
