> The solution here would be to have at least four (proven number to avoid oligopolies) private train providers competing on public infrastructure

It's mostly privately owned, but you are basically describing the current state of the current US rail network. The United States has an extremely extensive intercity rail network, with dozens of operators. This was basically all built by the dreaded free market, by the way.

The trouble is that the large railroads are only moving freight, because of the misguided decision to nationalize 20 passenger rail systems under Amtrak. We should have just continued to subsidize passenger service under the private operators.