So I'll take it that you do NOT have any examples of the private sector privatizing a function previously done by the government (in the US) and doing it better/cheaper.... Sounds to me like you've read some Ayn Rand books thought it sounded great, but haven't put forth the effort to imagine how a fictional tale translates to real life. I don't say this as someone who is pro-government. I'd love to see things done better/cheaper. I'm just someone who wants proof not promises...
No, I just grew up in a formerly communist country, and when you ask me for an example, the nation-sized experiment I brought up is the first thing that comes to my mind. You never specified you want to ask about US in particular, and I don’t know why you reject a clear and convincing example from outside US. In fact, I find your imagined version of me, with the Ayn Rand books and all, rather insulting.
In any case, there are plenty of examples in the US as well. Private education can successfully compete with public education, despite great structural disadvantages (private schools literally have to compete with free product). Using EBT at Walmart works better than government cheese. Section 8 housing works better than the projects. VA hospitals are not exactly preferred to regular ones by the public.
In general, in the US, the government doesn’t directly run too many things other than the regulatory agencies. Most of what it does is already outsourced. The US government doesn’t build infrastructure, doesn’t design or build military hardware etc., and most people (not just the straw man Ayn Rand fans in your head) would recoil in horror at the idea of government directly hiring people, buying materials, and managing operations.