I still don't see private corporations building better public infrastructure.
You get the quality of government you deserve. Remove corruption and regulate lobbying and donations and government money will be spent more effectively.
Between 1850 and 1872 extensive cessions of public lands were made to states and to railroad companies to promote railroad construction.[18] Usually the companies received from the federal government, in twenty- or fifty-mile strips, alternate sections of public land for each mile of track that was built. Responsibility for surveying and mapping the grants fell to the U.S. General Land Office, now the Bureau of Land Management. Numerous maps of the United States and individual states and counties were made which clearly indicated the sections of the granted land and the railroad rights-of-way.
"Still required" does not refute the fact that early railroads acquired immense real estate holdings which were (and remain) fantasically valueable as directly exploited, sold, or used as collaeteral in securing loans and bonds. The land itself immediately increased in value as transportation was made available to and through it. One excellent example is the city of Denver, Colorado, bypassed by the original transcontinenal railroad, and a sleepy town of 5,000 until a spur line was constructed to the Union Pacific's track. In a decade Denver grew tenfold.
A major cost, often the major cost, in new ground transport construction is land aquisition and accommodating conflicting uses. It's an irony of highway and railroad construction that it enables occupation and production of previously low-value land, but that the very increase in land values resulting transportation access makes additional transport construction far more expensive.
Another major cost factor is political opposition, quite often fuelled by competing transportation modalities. Today that's largely trucking and airline interests allied against rail, though the phenomenon isn't new, and as railroads were first emerging, it often came from exising canal companies and inerests.
See Bernhard J. Stern, "Resistances to the Adoption of Technological Innovations", 1937
In the United States when, in 1812, John Stevens wrote his Documents Tending to Prove the Superior Advantages of Railways and Steam-carriages over Canal Navigation addressed to the commissioners appointed by the State of New York to explore a route for the Erie Canal, his proposals were regarded as ingenious but visionary, and dismissal in the face of the recommendation for the canal by DeWitt Clinton, Gouverneur Morris, and Robert R. Livingston. The latter, Steven's brother-in-law, had been granted a monopoly to navigate the waters of New York State by steamboat and could, therefore, not be expected to be receptive. His letter, dated March 11, 1812, gives the reaction of an "expert" of the time...
And:
After New York had incurred a heavy debt in the construction of the Erie Canal, mass meetings throughout the State demanded that railroad competition should not be permitted to affect the receipts of the canal. When the charter of the Utica and Schenectady Railroad was granted in 1833, the line was prohibited from carrying any property except the baggage of passengers, a prohibition which prevailed until 1844, when permission to carry freight was granted but only when navigation was suspended and upon the payment of canal tolls.
Today, that role is often played by regional and budget airlines, rather famously, Southwest Airlines:
Texas Central Railway is trying to revive a part of that earlier project, a privately financed bullet train connecting Dallas and Houston. As the company prepares to do a federally required environmental impact study and hold public meetings along the planned route, its leaders say they expect to avoid the pitfalls of the earlier project, namely inadequate financing and intense opposition from Southwest Airlines.
The pattern of privately-funded ventures undertaking early infrastructure development, typically in low-cost, high-value regions, but transitioning to either a government-subsidised, or operated mode under expansion or as the market matures is in fact common. One such subsidy is often mail service (the withdrawal of postal sorting cars from passenger rail proved the final death knell, a case of death delivered by ZIP Code). For electrical and telephone service, you'll see this in both rural cooperatives, handling final distribution, and in major generation projects such as the Tennessee Valley and Bonneville Power authorities, and major dam projects (Hoover, Glen Canyon, Grand Coulee, and many others). Early privately-operated dams proved to have their own issues (see Johnstown). Many urban transit systems began life as private ventures, but came to be operated under government or quasi-governmental agencies in time (where the weren't destroyed outright through corporate conspiracy). Again, land-speculation and high-value routes are attractive to private operators. Serving high-need areas though wih little revenue potential, not so much.
> low-cost, high-value regions, but transitioning to either a government-subsidised, or operated mode under expansion or as the market matures
As you point out, the government's power is used to stamp out competition, or it subsidizes alternatives which drive the companies out of business.
> Serving high-need areas though wih little revenue potential
That's, by definition, not a high need. It's why our public transit is so expensive and such a small fraction of commute miles: because we don't have a focus on useful aka what people are willing to pay for. We let bums make it unsafe and dirty. We build it next to expensive, wide roads then wonder why people continue to drive when it doesn't actually go directly to where you want to go.
You get the quality of government you deserve. Remove corruption and regulate lobbying and donations and government money will be spent more effectively.