An entity that can rig the courts, the laws, and the regulations to its favor has an advantage.
Sure these are just anecdotes. But it puts a hole in the claim that the private sector is ALWAYS better.
Yeah, that's just magical thinking nonsense. You have to take the situation apart and look at what the incentives are. Economic libertarian woo is just as bad as alternative medicine woo. Markets aren't magic. They are a particular kind of distributed machine. It sounds like that mental hospital/home didn't have a competitive environment, or they couldn't have demanded more funding.
In fact almost every case I've seen where a private company does the job cheaper, it is because they give their employees worse conditions and salaries, not because of smarter organization and management.
In Washington state in the US, there were private DMV counters at Fred Meyer. AAA auto insurance can also do some of these functions. The customer experience is almost universally better at the private counters, because the private employees have incentives to make the customer experience pleasant, and the state employees have none. If a company can cheap out on its employees as compared to the state, and the level of service stays the same, then the market worked. If a company does that, and the level of service gets crappier, but the company doesn't face consequences, then the market has not worked. I bet, if you tried hard to prove the null hypothesis and looked for circumstances that would interfere with the market, you'd find them.