In respect of 'she's not a Tory, absolutely not!', curiously, during my most recent time away, I had a discussion with a moderate Tory voter which is relevant to this issue, feeding in as it does into Kendall's internal electability. They are well-informed, very middle of the road, totally un-tribal, a sort of David Willetts-style type. Couldn't really be called a Conservative, because they're not a member and would happily vote for another party should circumstances dictate.
We were discussing this, and he pointed out to me that Kendall struck him as decidedly Conservative-like at the start of her campaign, ('Even I was surprised' is the words he used IIRC) so it was usurping that she got the broad reception that she did. This kind of dumbfounded me a little, because to me Kendall it al are Blairites, and Blairites are not Tories. They're horrendously bad knock-offs of Tories. But I was left questioning to what extent my perception was defective, to what extent I'd been snared by a combination of my own political prejudices and repulsion at her treatment by social media.
I think people like Kendall are going to be victims of apparent association as long as the state of the Tories is what it is. Unless there is a shift away from the centre ground Labour modernisers will never have a distinctive political space to occupy which could be acceptable to Labour; but I still think, as I noted during the leadership campaign, that even if said space existed, said modernisers would also have to phrase that occupation of the centre ground in far, far different terms to that which they do, a distinctively Labour narrative.