This really is a bigger issue than gender ideology and the BBC.
And while it is also an issue across the right, I think it has a very peculiar twist across the progressive left spectrum because of the way that there is this real push to characterise other, be it other news sites, other institutions, other individuals, as purveyors of false news, while being totally blind to their the fact that they also have lost the public trust.
It's was a massive issue around covid where there were all kinds of accusations of people being anti-science while also resolutely avoiding any messaging from the scientific community that didn't support the correct "facts".
I have really noticed it on reddit - which is a pit but does give a window into the thinking of a certain group of people - with regard to the BBC treatment of Trump - flat out denial that it was possible followed by a denial that it was serious, and an almost total lack of understanding of normal standards of journalism, for decades, around things like splicing together footage.
I've seen similar with the reddit communities that follow the CBC here in Canada. Any criticism is shouted down as right wing people who hate the CBC and its spotless reputation. Even a book by one of the most venerable of CBC personalities was treated that way, as if the author was some right wing agitator out to destroy the national broadcaster. Not even recognising that no, this was the producer/host for decades on a flagship intellectual radio show, one of the people who created the reputation of the CBC in the first place. And no understanding why so man people who used to be listeners no longer trust them.
I've found it really difficult to understand where this kind of blindness comes from. But I do think it may be the same phenomena that has prevented left wing parties being able to respond to the public on issues they care about in recent years.
It reminds me most of the idea of prelest, in the Orthodox Church, which is a state of spiritual delusion that affects people who believe they are holier than they actually are. It's considered one of the most dangerous delusions because people will do so much evil when they believe they are doing so for righteous reasons, and they can often mislead others and destroy good institutions as well.