From what the poster I have quoted has said previously and from other things I've read, there can be a few issues in socially transitioning a child at school, particularly without their parents knowledge and consent.
There's been a steep rise in ROGD (Rapid Onset Gender Disphoria) where the child has never shown any signs whatsoever of questioning their gender previously and is what I believe rogdmum's user name refers to. The advice from medical professionals in this case is being ignored by the School.
There has been a massive rise particularly in girls with with ASD which may have less with truly questioning gender and more seeing things in a very defined way - 'I don't like pink- I must be a girl then as girls like pink'. Several posters in the past have described this experiences with their children. Again socially transitioning may not be the best option for those children.
Affirming at school but not at home (one name at school and one name at home) has to be confusing for the child surely?
The policy document links to various LGBT organisations who push for medical affirmation and suggests the School advises these as recommendations for the child.
And as an aside - the SG states several statistics in the document linked regarding suicide in transgender children which are untrue - they have admitted this themselves. They are used for 'illustrative purposes'. So the policies being used in Schools are incorrect and perpetuating a suicide myth which is harmful to children.