If an adult is similarly persuaded, she or he is generally regarded as a fool. The very idea that somebody's word counts as evidence rests on the assumption that they are competent to make their own decisions and answerable for them. To assume the same of a child is to take the child for an adult - the same mistake the abuser makes. In the seventies, before it was prohibited, the Paedophile Information Exchange used to argue that children were capable of making their own decisions about who they wanted to have sex with. Like Hillary Clinton, they argued that the child was competent.
Most people think otherwise.
When hearing the evidence of a child, any court takes that evidence for what it is - a child's word. A child can expect to be protected from cross examination because it is unreasonable to expect a child to defend itself. Any court would be bound to hear a child's evidence with a degree of scepticism, just as you would doubt a child that swore blind he had not put a stone through your window. When parents give them permission to make such decisions, they are educating them towards responsibility. But nobody should confuse permission with rights.