"I don't know at what point things like building roads became the responsibility of the government"
Transportation is meant to be utilitarian, and you can't just An-cap a solution to the problem. Freeways are easy to build, but extremely difficult to maintain without causing major productivity issues due to sheer size and importance.
A private company could build toll road and charge drivers for it. However, as time passes, maintenance is required. If they fail to maintain the road, it becomes basically a wasteland of blacktop no one wants to drive on. If they maintain the road, that's a whole lotta delays for many years.
The best part is that the state or another private company could take away customers from the toll road via building a far more profitable and lucrative *gasp* train service. Not only is said train service not exclusively to car drivers, it's also far easier to build and maintain.
As a result of toll roads failure to profit, especially with the malicious way toll road usage is charged, they maintain their status as a government utility. That hasn't stopped people from trying to profit off of them. Governor Ron DeSantis and the Floridian republican legislature have basically made every freeway in Florida into a toll road. When the company known as Brightline started planning out train tracks that paralleled Interstate 4 between Orlando and Tampa, the government wanted to have most if not all of the profit made from that service for every passenger taken off of I-4. Basically a hefty tax from a """"""pro-business"""""" Republican on a small company to help subsidize his shitty roads when he could be taxing billionaires.
"China is also an authoritarian state. Public transportation seems to be something all authoritarian states love to impose on the people and with good reason"
public transportation is not tied to the state of democracies in respective nations.
Autocratic nations with decent public transport:
China
Russia
Belarus(?)
Morocco
Autocratic nations with terrible public transport:
North Korea
United Arab Emirates
Egypt
Democracies/Free nations with terrible transit:
Canada
United States
Democracies/Free nations with decent transit:
United Kingdom
Netherlands
Japan (?)
"we don't have any free market examples other than the fact that private passenger rail all went out of business"
coincidentally, the government built freeways using railroad tax dollars, forcing them into bankruptcy. i don't think that's free market.