I'll grant you that People might know more TNG than DS9 or Voyager (And TNG was definitely more accessible to a wide range of people), but it wasn't a flagship show. When people think Star Trek, the first thing that comes to mind more often than not is Kirk and Spock. It was the original that started it all, and if it wasn't for the original, who knows what TNG would have been, if anything at all. What I didn't like was her shitting on the other series talking about TNG, and I can understand a lot of people being turned off by that. Like I said, TNG is my favorite series, and I was even turned off by it.
I'll agree with you on that when people think Trek, It's not Picard and Riker, it's Kirk and Spock. I think she meant modern Trek, in which case, it's a fair assessment.
I don't think it was meant to be taken that far, I really don't think that it went so far as to shit on the other series. She didn't state it well, I'll grant you, but to make the leap from she thinks TNG stands above the other series to that she has negative feelings about the other ones is a stretch in my opinion.