@moto748e You are assuming Streeting is only the messenger here, and the impetus has come from Starmer. How can we be sure about this? Based on both men's past statements, it seems much more likely to me that this was WS's' own initiative, which actually might embarrass or disconcert Starmer a bit.
My comment was interpretive (and a bit overwrought, sometimes I can't help myself). But I do think that it's important to put this particular issue into a broader context. Many have argued that it was the reason Harris lost and Trump won. I think that's correct. The big dynamic going on in UK and US politics is the collapse in credibility of civic and governing institutions. The reason why the trans issue has become salient as a national issue is because it's an ideology so insane that it could only have been rolled out by institutions that are immune to popular opinion and the pressure of politicians that need to win democratic elections.
In this way the issue is deadly to center left politics whose politicians win elections by claiming to be better administrators of government, and by promising to use the power of government to provide services that improve peoples lives. Reform/Trumpism argues the institutions used to deliver those services are hopelessly corrupt and should instead be dismantled and the proceeds used (for example) to fund tax cuts.
I think there's evidence that Starmer grasps this and in fact it's his principle political concern. Starmer himself is probably not terribly committed to the GC cause out of principle, though he's probably miffed that it's caused him personal embarrassment. Streeting clearly is however. And i think it's reasonable to assume that that's one of the reasons he was selected for his post. It's also notable that there is not a single TA in government. Saying that Streeting was dispatched was bit of a rhetorical flourish, but surely Starmer is aware and approves.
In any case, I do think the situation that I've sketched is consistent with how Starmer's actions are playing out more broadly, but you're quite right that it's speculative. And also quite right to be distrustful that Starmer is GC convert.