Both issues have a lot in common, but not in the way that you think.
There is a very similar pattern here of activists and party apparatchiks on the right, creating a shitstorm of bogus allegations in order to demonise figures in the party they want to get rid of. It is all about delegitimising narratives which conflict with the ruling class ideology. The more the ruling class ideology conflicts with reality, the harder the attack against dissent has to be.
The movers and shakers behind the Labour smears, the “Campaign Against Antisemitism” and “Jewish Labour Movement” are misnamed: they are not a campaign against antisemitism or a representation organisation of Jewish workers, they are hardline pro-Israel lobby groups, and they pursue their lobby in the same way as Stonewall and Scottish Trans Alliance do – with lies, exaggeration, DARVO, dogpiles and denunciation of “bigots”.
Political arguments are de-politicised and reframed to make them about “hate”: “You hate trans people” or “You hate Jews”. The intended effect is to isolate – who wants to defend a “bigot”?
The escalation of every dispute is deliberate. The purpose is firstly to make people think there is no smoke without fire, and secondly to deter others from getting involved in the discussion, because you will think twice about speaking out if you are immediately subjected to all sorts.
The same tactics of smears and lies are used to make you accept the occupation of the Palestinian territories, and to make you pretend you believe men can be women.
That's my analysis. I don't expect everyone to agree. But mine leads me to think it is far more likely that Starmer will attack Duffield than that he will suddenly spring to her defence.