I was responding (originally) to the article, which I find to be the kind of source that could be used to claim anything about anyone. The linked article is based off an article in The Daily Mail, a famously tabloidy, unreliable, and awful British newspaper. They enjoy shit-stirring and generally making anything look like a conflict. The Daily Mail article was writing about a paywall'd interview, which the blog author admits to not reading. So this is secondhand filtered through a very unreliable source (and if you check the original Daily Mail article, it looks like the writer thinks the more scandalous part is Chelsea affirming women can have a penis). And - even given that - the blog author is still spinning the comments in a very specific way. I don't think it's clear at all that Clinton meant these statements at all in the way the writer took them. Certainly, if you take away the nameplate and have it be solely comments from Person XYZ - there's nothing shocking here. "Old people are bigots", and it's entirely reasonable that the speaker is calling for "sensitivity" for trans people. Or that calling for sensitivity for old bigots just means telling them they're wrong politely.
Essentially, a source like that is too weak. If you are already inclined to distrust the speaker, then sure, it confirms your dark suspicions. If you are inclined to trust the speaker, this means nothing, since it's essentially reading a coded message in that goes counter to what everything else in this interview and this book are saying. What raised my eyebrows originally was the fact that the article itself, and I presumed CK at first (but I was clearly wrong on this), took it as a switcheroo - that the speaker was good, but based on a weird coded interpretation of a phrase, now they're bad. In my defense, this is the slant the article itself has! It notes Clinton's strong history on the subject recently. But if such claims were accepted as reasonable for any sort of change in opinion, that's just a formula to get played by Russian trolls. If Fox News runs a story saying "Bernie Sanders caught admitting that capitalism is awesome, abandons socialist principles in interview," just accepting this as true isn't wise, even if Fox dug up some quotes that might indicate this. (And yeah, Daily Mail & Fox News are about the same level.)
Now, to toss this all aside for a moment - obviously, there were names in the article, not a talk about Mysterious Person XYZ. So, if you already do not trust Hillary Clinton on this issue, if you previously thought she was TERFy or the like, I actually don't have too much to say, that's a different discussion (and again, this was not obvious by the article itself). If we start with the hostile outlook, it's like if an old-school Southern politician who supports actually racist policies is caught saying something maybe explicitly racist: if it confirms what everybody suspected anyway, sure, no benefit of the doubt. (Although, even then, I'd argue the "gotcha" gaffe is less important than the actually bad policies to begin with.) I'm merely saying that from a neutral or positive starting perspective - it's, at best, not convincing this was actually a change in policy to begin with, that this was meant as a retreat from trans rights or anything like that. It is just an indication that the interviewer asked about the topic.
I will say that the Clintons (Chelsea, but endorsed by her mom) explicitly affirmed that trans women are women in the interview, though, with regard to your other comments, and did not condition this on finishing a medical transition.
