I would think the main reason a child wouldn't want to disclose their gender distress at home is because they don't feel safe or confident of parental support in doing so. For that reason I think the school are correct in their policy and considering the safety of the child.
Ultimately, if a child asks a school not to disclose, I think that the school should treat that as a huge red flag about the parents and disclose that information to social services at the earliest opportunity so they can investigate why that child feels like that.
Maybe social transition is an indicator of medical transition but, what we do know is that if around 10,000 children have been referred to GIDS over recent years out of a pool of around 12.5 million under 18 's around 99.92% of under 18's in the UK are never referred to adolescent gender services. What we also know is that even if children are referred they will need parental/guardian/social services support for any interventions. This is a matter of law as we know from the recent court case that made much of the Bell judgement around consent irrelevant for those with supportive parents. Further we know that referrals will wait around three years to be seen. Add in a likely further 18 months to two years of assessments by GIDS and UCH we can with some level of confidence say that anyone referred beyond the age of 13 today will be unlikely to receive any interventions on the NHS before adulthood. Based on the current estimates banded around of between 0.3% and 1% of the population being trans, this number referrred is smaller than I'd expect. Even of those referred, I'm led to believe only 20% or less go beyond a consultation stage. Enter stage right the much misrepresented 80% desistence statistic that is often misapplied. So what we have in 0.08% of under 18's being referred to GIDS of which 80 or so % to no further than chatting about their feelings before realising they're probably not trans anyway. The group that is left, those who are determined to be persistent, insistent and consistent really don't deserve the hostility that directed towards them and those who support them.
All that leads me to think that we live in a society where a large number of trans people surpress their distress and identity until adulthood. After all, if there are a few hundred thousand trans adults why do few trans children? There could be many reasons for that but an expectation of unsupportive parents or even a fear of abuse and transphobia can't be discounted as a significant factor and is a very real concern for this tiny cohort of children. Many trans people are likely to support this idea given their experiences.
Given all that, the small numbers involved, the likelihood of adolescent treatment even occurring and the absolutely undisputed fact that there are abusive parents out there, I think schools would be neglectful and playing fast and loose with the safety of their pupils if they had a blanket policy of always disclosing their support for a child's sense of self to their parents. Of course in many cases it won't be a problem but, the safeguarding risk of always doing so is in my opinion huge and one schools can't afford to dismiss even at the behest of pressure from the Safe Schools Alliance of Transgender Trend.