Actually I'm under the impression that the ‘standard’ resolution is not about the “harm” part but about the “want” part

I think your impression is mistaken.

it's assumed that people below a certain age can't want sex, to the point that said age is called the age of consent

Nope. It is assumed that people below a certain age cannot give informed consent. In other words, they are assumed to be not capable of good decisions and to be not responsible for the consequences. What they want is irrelevant. If you're below the appropriate age of consent, you cannot sign a valid contract, for example.

Below the age of consent you basically lack the legal capacity to agree to something.