I think they don't like feminism because, as was alluded to in the interview, they see it as some sort of personal criticism. Like when they were discussing prostitution at the end and Konstantin was saying that the message would hit home more with him if it was directed at 'men who pay for sex' rather than men as a whole. It's a bit NAMALT (and please recognise me as a good guy), which ignores the fact that it is about whole society attitudes and if the 'good guys' were vocally opposed and held their peers to account, maybe fewer men would feel entitled to women's bodies and labour.
He did rather hone in on the firefighter issue and it was a bit distracting, but because Julie had already said she wasn't that interested in those sorts of issues, she wasn't going to explain the wider sexism issue, which is that yes, in the main, firefighters are men and traditionally it was only men, but now there are women, someone really should have recognised that women are not just small men, but a completely different shape, and designed something safe for them. See also: police stab vests. That stuff does matter and the fact that it is overlooked is sexist. No, it's not rampant misogyny, but taken as a whole with all of the other similar 'man as default human' examples, it is sexist and does matter.
They bought into Julie's argument that the major issues are the massive, life or death ones like rape, femicide, domestic violence...etc, but I believe that sexism in a wider and general sense does have a role in those issues too. All of it is about seeing women as lesser. It does seem, though, that they would see the smaller issues as 'whinging' and irritating - which I think is partly to do with the fact that, if we take the smaller things seriously, they (the good guys) would need to look at some of their behaviour too and they don't really see why they should have to (being the good guys).
Just thinking aloud...