What you describe already occurs. Would not legalisation of sexual relations with children remove much of the potential customer base for said prostitution? Why would I (just as an example) take part in such things when I could just ask the parents for permission to court their daughter legally? If they ask for money or other illegal criteria, I would then instead go elsewhere.
Also, I'm certain many parents would disagree with your assertion that children are "easy to control"! I can also say with some authority (as an old carny and stage magician) that adults are by far easier to manipulate than children. Children are much more inquisitive and ask questions where adults fear to "show their ignorance".
• ( https link ) ALICE DAY