You can straightforwardly privatize things that can't be allowed to fail. Power generation is a decent example: If you have a hundred independent generating companies that all sell into a live auction market, they have to compete with each other which keeps prices down and the market as a whole remains reliable because there are many independent suppliers using different generation methods and if any one of them experiences a failure you can immediately switch to the others. The market has the incentive to be able to supply even during grid stress because the price will go up then, so suppliers with reserve capacity make back their costs for keeping it in reserve.
The problem comes when there is corruption. Instead of having a competitive market, the contract is assigned to cronies, or a lack of antitrust enforcement allows the market to consolidate until the suppliers are a cartel. Then, since competition is thing that gives privatization its advantage, the lack of competition causes the public to get screwed.