No one has ever said that private businesses are "perfect", but competition literally leads to excellence everywhere. Eventually, the cheapest ones, the fastest ones, the most reliable ones will ALWAYS stand out, making their customers' lives a lot easier. This is not rocket science.
What happens in Brazil, though, is that the government knows it as well as we do. And since those bureaucrats live off the people's taxes, one can never expect that they'll give free market a go, otherwise state-owned institutions like Correios would certainly cease to exist.