The idea that boys 'should' be like this and girls 'should' be like that is precisely the kind of gender stereotyping that gender critical people are critical of.
How does the charity define a 'transgender child' if not a boy who isn't doing boyness right, or a girl who isn't doing girlness right? Therefore they must be the wrong sex/in the wrong body.
The whole concept of 'a transgender child' is based on gender stereotypes, and putting that inappropriate, contested adult label on a child is detrimental.
The mere fact that the charity in question uses the term 'transgender child' - and rejects any child who might not go along with it - makes a statement, and reveals an unhelpful attitude to gender-questioning children.
I feel sorry for the child who was rejected because of his mother's opinions, but I also feel sorry for the 'transgender child' who, instead of being given the support they need to accept and love themselves as they are, will instead have an adult-defined disconnect with biological reality reinforced.