Sometimes libertarians are operating in the realm of sheer fantasy with no actual real-world examples of the magical benefits they claim will happen out of their various privatization schemes, but in this case there is a contemporary example of a major city that largely left the free market to determine its surface local transit, Seoul, South Korea prior to 2004:
The likely libertarian excuse for all this will be the government controlling the fares and providing operating subsidies - of course the fact that the government found it necessary to control the fares to keep transit affordable to the people most in need of it, the working class & poor won't register with them. Instead the price controls & subsidies will be some do-gooder activist government intervening apropo of nothing, lacking adequate faith that eventually the free market will cure all its own self created ills.
Nonetheless, even if we accept that fare controls & subsidies have some negative effects, the multiplicity of coordination problems and unethical competitive behaviour are inescapably the consequences of running transit as a for-profit business. Each transit company still competes for passengers and has incentive to overcrowd, rush & avoid unprofitable passengers.
All of this goes into why in 2004 Seoul found it necessary to greatly extend the level of government control over the surface transit system, taking control over schedules & routes in addition to fares (while leaving the private companies to still operate the system).
Some context about Seoul is important here - today it is a (groaning as I type this overused moniker) "World Class City" and biggest city in a prosperous rich-world democracy, but that is quite a recent development. South Korea's income per capita rose tremendously in recent decades so the fact that its bus system was very nearly fully private is an artifact of a developing world governance structure & capability, it also takes place in the context of a society that mostly couldn't afford cars, and thus most people had no other options than transit, however unsafe & inefficient it was. Once prosperity rose to the level of mass car attainability, use of this wild west bus system fell precipitiously (ibid):
The alternative is to realize that transit is a natural government function as roads & rails are natural monopolies - there's only so much land to build them on, and use is rivalrous, so each competitor on the network makes the network worse. Government must govern these things if they are to work for the people who need them (rather than for shareholders). The actual drivers & operators may not need be government employees but the system as a whole needs to be, well, centrally planned or it will be a disaster. Public transit is a utility, a necessary and proper function of government.
This is really quite common for libertarian fantasies - you usually find that their fantasy system used to exist in the early stages of societal development and the government intrusive/managed systems were introduced to solve the obvious, real and pervasive problems that leaving whatever function to the free market created. Liberal do-gooders and socialists don't win these arguments to have government intervene unless there are already big problems with the status quo. That government runs transit in most places is no different from why every rich world society found it necessary to have government manage health care, the US being just the latest (and very late) example: If you want these things to work, government must be involved.
As usual libertarians, it has been tried your way, and it failed, which is why the government is in there to begin with.
Incidentally the article I'm quoting is an excellent look at Seoul's stupendous transit system, and more importantly how it got that way, as opposed to "transit porn" pictures of the subway route maps without any context of how it was all funded and built.