Ones of the reasons used to oppose private public transit is that bus and especially rail routes are natural monopolies and so leaving those services to the free market would be tantamount to granting a company a monopoly on transit, resulting in a rentier situation where the company would never have to improve its service or find savings in its operations to make a profit.
Other concerns include the ideas of equal access — ensuring that poor and/or minority people have the same access to transit as wealthier, whiter people. However, equal access can be accomplished in multiple ways by municipal governments without being the responsibility of the transit agency.
However, I think that the free market can provide transit in cities without any problems with monopolies (or access). The problem comes in for two reasons: one, even when built, owned and operated by a for-profit company, transit service was typically tied to the local or state government through franchises. In New York City in the 19th century, for example, a private firm could not simply buy land or air rights from the existing owners. They had to buy a franchise from the City and get their corporation chartered by the state. The franchises were essentially a way of rewarding businessmen and politicians with good connections, similar to the way the transcontinental railroad got land from the federal government as it went west.
The artificial restriction on possible routes probably actually resulted in lower costs for the early rail systems (since they wouldn’t have to buy air or land rights from exisating property owners), but severely restricted competition. The homestead principle, as articulated by Murray Rothbard and as practiced in radio before the Federal Communications Commission was created, would be much better for private mass transit.
Before the FCC was created radio stations in an area simply took the bandwidth available to them — if two radio stations in the same broadcasting area attempted to operate on the same bandwidth, courts ruled that the one that started earlier was the owner. In other words, bandwidth was distributed on a first come, first serve basis. After the FCC, the federal government effectively became the owner of the radio spectrum in the United States and auctioned off bandwidths to the highest bidder, similar to NYC’s transit franchises. This raised the costs of starting a radio station and limited competition.
With homesteading, a prospective railroad might acquire the most direct geographic routes in a city first, but the most direct route isn’t neccesarily the most profitable route. While some customers might prefer a more direct route, I think most people would prefer a more convenient one. I know I don’t like walking more than a block or two to get to a bus stop or train station and I might just pay a premium to use a more convenient one. Homesteading routes would also allow a prospective metro system to avoid following streets by allowing the company to buy the property it would need for its routes.
Under the transit agency monopoly paradigm, things aren’t much better than in a rentier situation. In fact, with many transit agencies dependent upon annual bailouts to pay their operating costs and debt service, while deferring maintaince and funding contracts, it could be argued that the existing situation is worse in almost every respect compared to a private firm monopoly — the private firm could shut down its services if it wasn’t making a profit, while there’s an expectation that the transit agency would continue operating. Although this is not entirely clear — if states choose not to bail them out and their level of indebtedness prevents them from borrowing more, service would probably shut down.
Equal access would probably be improved with for-profit transit because it would be in low-income communities, where people are less likely to have access to alternative forms of transportation, like cars, that a transit company would have its most dedicated revenue stream. Currently, especially with newer rail systems, the revenue potential of serving low-income neighborhoods is often sacrificed by transit agencies that are pressured into serving political ends, since the politicians control the funds.
For example, new rail sysytems are mostly downtown light rail, which combine the disadvantages of subways and buses, but none of the advantages: like subways they’re expensive to build and operate, but like buses they have a small passenger capacity and can get stuck in traffic. Everyone knows this about light rail, but the politicians and bureaucrats are willing to subsidize it anyways because they look at old photographs that show streets active with people, filled storefronts and streetcars and so they think that to get 1) and 2) they just need to bring back 3). The old pictures don’t show highways and parking lots, which are the chief causes of urban grief.