The idea that trans women have anything against lesbians is absurd on its face. Anyone who spends any time in real LGBTQ spaces and communities knows that is not true.
It’s equally not true that lesbians in general have anything against transgender people.
A few loud, intensely trans-hating lesbians take up a lot of media space, but they in no way represent lesbians at large. From what I can see, they appear to have some intense, deeply personally rooted problem that fuels their public hatred.
All the lesbians I know (and I know very many lesbians) have no problem with trans people. Their attitudes range from uninterested, to supportively neutral, to aggressively affirming. I’ve seen a lot of data that back up my personal observations.
It’s really astonishing that a tiny handful of trans-hating lesbians, who clearly have psychological problems and who’ve gone out of their way to create organizations to hate publicly on trans people, get treated by the press like they are community representatives.
That treatment is symptomatic of wider transphobia, of course, as you point out. When the press latch on to clearly flawed research and an unrepresentative smattering of mentally ill haters, you know they are doing it because they have an agenda.