Yes, no one has to push, demand or coerce if it's going to happen without any of that. The children involved, in my experience, are not the pushy, demanding and coercive types. They are more the quiet, vulnerable, more likely to be bullied types. People, particularly teachers but also other 'nice' children are inclined to feel protective of them, and want to do what they can to make their lives a little easier. If this is calling them by a preferred name and pronouns, then this is what happens (note: this happens even without a school instruction, lots of kids socially transition among their friends first before the school gets on board).
And people can say Cass says no, but you need to remember that the Cass review was only published two years ago and schools have been dealing with this for well over a decade. And for well over a decade the understanding has been that deliberate misgendering is a dick move.
Consider the technical guidance to schools from the EHRC that was published in 2014 which contains:
"A member of school staff repeatedly tells a transsexual pupil that ‘he’ should not dress like a girl and that ‘he’ looks silly, which causes the pupil great distress. This would not be covered by the harassment provisions, because it is related to gender reassignment, but could constitute direct discrimination on the grounds of gender reassignment."
You can tell how long ago it was written (it's still live) by the fact that it uses 'transsexual'. Note that the example for direct discrimination on the grounds of gender reassignment explicitly uses the misgendering 'he' in its example of a teacher who is potentially breaking the law.
Don't blame schools for socially transitioning kids when that's the legal guidance we've been given, that the DfE have explicitly directed us to teach about gender identity as fact. It hasn't required pushing or demanding or activist teachers, just reading the documents we've been provided and acting on them.