Great video. Although the Uncharted bit was silly, Naughty Dog always said they start from set-pieces/gameplay mechanics and build the story around them. (And they are pretty successful lol)
I don't think they ever said that about gameplay, only set pieces. It definitely does show, especially in Uncharted. The jokes about Nathan Drake being a serial killer may just be jokes but they show that the story and moment-to-moment gameplay are not fully in-sync. In UC4 for example they added further stealth elements just because they were coming off TLoU and more and more action games are using stealth. I never got the sense that it influenced the story or really many other parts about the game. If Naughty Dog was taking a Nintendo approach with The Last of Us then Ellie and Joel's interactions would be the entire core of the game instead of just a very small part of it.