I understand your point, but am providing a real world example in the form of a privately run railroad that was more efficient than its subsidized competitors.
> put the burden of paying for that infrastructure entirely on people who travel instead of all the people who benefit thereby
I agree with this part of the statement, but it's false to assume the free market can't utilize this payment model.
The bulk of the railways and roads that I cited were paid for not only with user fees but investments by businesses or entrepreneurs who rightly assumed that they'd reap benefits from railways and roads being constraucted in their districts.
In the case of the Great Northern railroad, the efficiencies were passed on to the customer in terms of ever falling costs and an expanding network of service.
> The population of low-volume regions will be under-served
No, it's the opposite! Low-volume regions are underserved with centrally planned government transportation. The planner has limited resources that must be "democratically" administered.
In a free market, any underserved area provides an opportunity for an entrepreneur to serve that need in the market! It's an elegant system that serves the needs of the people faster with less waste.