It is possible to understand and accept two positions simultaneously.
The first is that the labour anti-semitism "controversy" was largely manufactured by a charity which mysteriously popped into existence shortly before the 2015 general election (and seems suspiciously more concerned with finding anti-semitism in the labour party than with growing visibility and violence of the far right in recent years) and that these claims were siezed upon by the largely right-wing British media as a deliberate way to attack Corbyn's leadership.
The second is that there is a minority among the left of the labour party who do not seem to have an adequate understanding of the line between anti-Zionism and anti-semitism, who use terms like Zionism in ways that are ambiguous and conspiratorial and in some cases do this because they are antisemites who do actually believe in a global Jewish conspiracy, not because they are poor at choosing words.
Likewise, it is possible to believe simultaneously that Corbyn's media strategy over the issue was shit, and that the British media was so virulently opposed to him that no media strategy would have been good enough.
Living through the whole trans panic in the UK over the past few years has put a lot of things into perspective for me. In a way it has made me more sympathetic to the position of the CAA and its supporters. Seeing harmful discourses circulate in public space can be far more painful than being at risk of actual violence or harassment, even though the latter is undeniably worse. On the other hand, it has made me very conscious of the fact that the kind of bigotry the CAA claims is rife within the labour party is actually just everywhere in British society, and isn't met with anything near the same level of (admittedly fake) sympathy.
Imagine if we got together and produced an "international definition of transphobia" that included things like using the term "trans ideology" in an insufficiently nuanced way, using AIDS-panic stereotypes like trans people being a threat to children or using any language that might imply trans people are mentally ill or less reasonable than cis people. These are all valid concerns rooted in the particular history of bigotry against queer people in general and trans people in particular, and yet noone is going to care because if they did half the Tory MPs and journalists in this country would need to resign.
Of course, the CAA isn't responsible for the way their statements get used since their job is just to point out antisemitism, but it isn't fair, and exploiting a position of unfairness doesn't build much sympathy with me.