Gender vs. Sex

So we've established that gender and biological sex are two different things. Gender is based on cultural values while sex is which of three bodies you are born into (male,female,intersex). My question is, what does someone think when they identify as transgender? If you were born with a penis, but like to wear dresses and makeup how does that equate to also feeling like you should have been born with a vagina? If we get rid of all gender roles and stereotypes then what makes someone transgender? The fact that they want different genital parts then the ones they have? 
It troubles me the most when talking about children and allowing children to transition has been a very hot topic. I'm all for allowing children to express themselves without imposing gender roles but why on earth would you tell a child it's totally normal to be distressed over your biological sex and it's okay to hate your genitals and wish they were different. It's the only thing well allow children to hate about themselves and actually validate that they should change. If your child hated their nose, their height, their weight, their legs or anything else about themselves you would encourage them to form a new perspective. And yet when they hate their genitals all the sudden it's okay and validated?