Actually I don't see that it necessarily would have been changed on SIMs etc unless it was an official legal name change. You're not supposed to change the child's legal details without legal reason to do so.
For (very common) example, mum shacks up with a new boyfriend and tries to change child's last name to boyfriend's name. We didn't change names without legal documentation in any school I ever worked in. That's why you'd usually go down the register in a new class and get a handful of kids telling you their preferred names. It's entirely up to the teacher whether they pander to that or not, and usually teachers (being well educated adults) can use their own judgement.
I'd also counter argue the "school isn't the place for teachers to share their personal beliefs" that this should apply to teachers who buy into gender woo as well, so going along with trans names and pronouns is also sharing their personal (pro gender) beliefs. 🤷♀️ There is no way of getting that right that works for everyone at the moment which is a big issue for why that trans guidance was so needed.