Clearly a local government is going to be less receptive to more digging if high speed service is in place, but it's not a "natural" monopoly if a deal with the government must be made as a prerequisite to provide services. You are also assuming that the "digger" is necessarily the service provider. In my view, local governments are basically seduced by telcos subsidizing or eliminating the up front costs of building infrastructure. In the long run, it's a bad deal.

I would say the monopoly is "natural" if and only if the local government is required to allow anyone (or no one) to dig. Or if it pays for the digging itself.