No, don't have a monopoly run it. But also don't require every ISP to lay their own fiber. Do what they do in Sweden. And no, it is not some communist/socialist monopoly. They have a single entity lay the fiber, but then let many, commercial, ISPs compete to provide the service over the fibers. It works great and is less expensive than what we have here. And Sweden is not that densely populated.
Yes, I am worried about the entity (whether it is gov't or a regulated commercial entity) that lays the fibers getting out of hand with their tariffs, but overall, it would seem to provide the best opportunity to get the US out of third world status when it comes to internet access.