A few days ago, CNN listed Hong Kong, Seoul, Singapore, London, Paris, Madrid, NYC, Tokyo, and Guangzhou as being in the top list of subway systems. At least some of those were started either by the government or by a government owned/funded company.
The US has a long history of preferring business solutions over government ones. Compare to 1800s France where large-scale engineering projects were seen as part of the government's domain. I therefore suspect that your observation reflects a US-centric view, where the default view seems to be that the government is incapable.
Also, there's a possible selection error. If all of the worst systems were "originally built by private companies" and only 40% of the world-class systems were done that way, then wouldn't that suggest that early government involvement leads to better systems?
Without that information, it's very hard to use your observation to make any sort of informed decision.
I suspect the same is true for your other examples as well.. how do you "private sector" build something like a subway? The closest you could conceivably get would still be quasi-public, once you get done with all the right-of-way and eminent domain concerns.
BTW, you "private-sector" build a subway the same way you do an oil pipeline or a transcontinental railway (eg, Hill's Great Northern Line http://en.wikipedia.org/wiki/Great_Northern_Railway_%28U.S.%... )
I wouldn't call the CTA Rail world class. Compared to NYC or London it has limited stops and doesn't touch large chunks of the city.
CTA: http://www.transitchicago.com/assets/1/maps/P19_2012_CTA_Rai...
Tube:
http://www.tfl.gov.uk/assets/downloads/standard-tube-map.gif
Chicago's bus system doesn't really make up for the lack of stops. Lots of people consider a car mandatory here, unlike London or NYC.
The parts of the city that aren't directly served by train are served by a relatively effective bus system. I wouldn't say the CTA buses are beloved of anyone, but they're night and day compared to SF.
The public ends up funding projects that might be valuable to society, but that won't turn a profit. Of course, this tendency needs to be monitored to make sure that the projects actually are of value, but I would say that public transit in San Francisco-- if nothing else, it brings down real estate prices-- is clearly in that category.
I do not intend to say that public projects are always well-managed, because that's demonstrably false. However, there are a large number of badly-managed private projects as well. In terms of turning a profit, though, the private sector almost always has better material to work with, because it takes the projects that are visibly likely to be profitable.