As a former local authority PSHE Advisor I recommend you have a whole school policy on this. The very fact that your PSHE coordinator is scratching around and randomly choosing resources is very concerning. SLT should be aware of what's happening and funding appropriate training and resources. What if you hadn't overheard the initial conversation? It really should not be down to chance. You sound very experienced and handled the situation well.
To be fair, he wasn’t randomly assigning resources. He couldn’t find any that adequately explain transgender (under the equality act), gender identity (under the RSE statutory guidance) and avoided gender stereotypes (which we are required to counteract). The usually good sources are shockingly bad on this topic (eg PSHE Association). I have raised issues in the past (re the Brooke Advisory) and we got them changed, plus did clarification lessons.
School staff are human. We make mistakes. On this issue we cannot rely on sources we should be able to trust (eg stonewalled NHS, contradictory government guidance). Of course it shouldn’t be down to luck, or chance. And it isn’t entirely, because our biology curriculum is explicitly clear on what sex is and that it cannot be changed. There is definitely a way to go, but unless or until schools get properly funded, assessed, and resourced with respect to PSHE we will be reliant on individual staff noticing an inconsistency.
I do think I handled it well though (this thread really helped). I’m gonna take that as a win for today.
PS, I don’t think it is possible to teach gender identity and to counteract gender stereotypes, so I think it is incredibly unfair that teachers are being expected to do both at the same time. Until the government gets a flipping grip we’ll keep doing the best we can to meet our legal obligations.