The analogy with a river doesn't work at all. Rivers don't act or engage in exchange. I see what you are saying on the most optimal path for water to flow, but that is then equivocating on "monopoly".
The lowest energy state of the natural monopoly is a single firm. Excited states can also exist, and are necessary for transitions to occur, but these states are less economically efficient. Customers see higher prices, and firms see lower profits. But given that a monopoly is in itself inefficient, it can still be worthwhile for a challenger to attempt to displace the incumbent, especially if they can operate at lower costs.
In a truly free market, the switch could be accomplished by an initial investment of sufficient size. In the context of the analogy, huge machines dredge a new channel for the river. For a time, water flows in both channels, but in the end, one dries up and you are left with one river again.