I was the victim of horrific online harassment from one poster on another website many years ago. She seemed to absolutely loathe me and would post things like 'just do everyone a favour and kill yourself' when I would post about feeling suicidal. So I do know the impact vicious communication can have on a person, but on the other hand, I do get the points posted recently about freedom of speech.
That is a good example where it would be completely reasonable to go to the police and complain about harassment. That easily reaches the threshold of persistently causing annoyance and anxiety as stated in the 2003 Communications Act.
Nothing Kate wrote approached that level. It was mostly stuff like misgendering and using terms like "pig in a wig". A lot of the time she'd blocked Hayden's account so it was, as it were, behind Hayden's back.
People write stuff like that on Twitter all the time. As far as Kate was concerned, she was just engaged in the usual exchange of insults and name-calling that is pretty routine on the platform. It took someone like Hayden, a persistent and determined litigant, to seize the opportunity to report that as a criminal offence.