They both (Contrapoints more so than Noah) seem to think that Rowling did not arrive at her views by rational means and that the only correct (rational) position for Rowling would be for her to change her mind and agree with them.
I suspect this is because they value feelings over rational thought when arriving at their own conclusions. They therefore assume JKR has done the same.
There was an interesting programme on R4 a few weeks back about the idea of a cultural shift away from objective truth to 'feelings', and the concept of 'my truth' in society. A couple of students talk about how feelings are the foundation of all their thought.
Many young people feel JKR must be transphobic to have said anything that threatens to burst the all-important TWAW TMAM bubble, so they seek facts to prove her transphobia. They haven't found any in her actual words so they conclude (because they've already made their minds up) that she must simply be motivated to say non-transphobic things from a starting point of unacknowledged transphobia.
I've seen the same thing on other FWR threads about Kelly Jay Keen. Posters who say 'KJK hates trans people' but who are unable to articulate exactly what she's said that proves this. Their answer seems to be that the hate-filled subtext is 'obvious'.
CP and Noah are saying the same thing, albeit in milder terms. JKR hasn't actually said anything transphobic but I still feel she must have misguided transphobic motivations and if I talked to her about how it makes us feel, her feelings will change and then she will think differently.