I would suggest reading Alon Levy's blog, where he writes in great detail about the (mis)management of public works projects related to transit.
Pedestrian Observations | For Walkability and Good Transit, and Against Boondoggles and Pollution
He provides a lot of details about why these sort of projects in the "Anglosphere" end up costing too much and being poorly designed, but he does show plenty of public sector projects in other parts of the world get built efficiently. One of his suggestions is that US transit and rail agencies hire executives from Spain or Italy, or Asia, but I think that even then, those execs would be frustrated by American political culture.
As for private capital being able to build things on time and under budget, that's not always a good thing, as readers of
John Howard Kunstler's books about sprawl would reveal. (Note: Kunstler has more recently slipped into conspiracy-mongering, anti-vaxxer crankery, but his stuff from the 1990s and early 2000s is still good.) Yeah, they built all those housing developments, office parks, malls, etc., and made a profit, but the stuff they built was (and is) junk, cluttering up our landscape and being the main source of the climate change apocalypse that will overtake us sooner or later. Of course, all that "private sector" suburban build out was enabled by massive government highway building projects that had all of the same types of project mismanagement we see on rail projects.
Anyway, all that private-sector sprawl buildout was done on greenfield sites, so no wonder they could be done on-time and under budget. Private capital usually skims off the cream for their projects, they don't touch anything that's going to be too complicated. So of course, they appear more "efficient." So what? If private capital could run passenger rail better, they would not have offloaded the service to the public sector. And don't go yammering about Brightline. That is a very special case. And I am still putting even money on Brightline's owners trying to offload the train service to the public sector at some time in the future once they make their money from the real estate deals.