Minaeve isn't Dalish, she was only born there. And Vivienne changed the subject too quickly for me to speak more (it also wasn't an unfinished sentence, and Minaeve doesn't contradict the Inquisitor on that note--the line you mentioned is from Vivienne only).
"You aren't a real Dalish, you only lived there and was raised into that culture."-is by far one of the most idiotic arguments I have ever heard. Vivienne didn't change the subject too quickly, the Inquisitor has no response, though I'm sure you'll pull one from somewhere. As to Minaeve, she makes it clear what she knows Dalish to do. If anything, she makes the Inquisitor's clan the outlier, and from all sources we have on it, it is. Even then, this clan does not keep the extra mage children. Tell me, where will they go then? When all other possibilities are removed, there is only one option.
Well, there's one Dalish who's a threat to her.
"The Inquisitor isn't Dalish, they just grew up there, but now they live in Skyhold, so they aren't Dalish." And even then, the observation has no affect on her whatsoever. You just don't want to admit that even "the People" fear magic and would rather kill extra mages than allow them to join the Circle.