So chiming in on this dumb argument because hey this is the internet and this is NeoGAF, I personally have no inherent interest in Naughty Dog taking the Half-Life mantle simply because I don't feel they've demonstrated many franchise specific qualities that for me define Half-Life and would thus make them ideal candidates to continue the series in Valve's absence. It's not a slight against Naughty Dog's talents, especially in the narrative department, but that their best work in Uncharted and The Last of Us strikes me as a distinctly different style of game design from top-to-bottom of which isn't necessarily transferable to Half-Life. Which doesn't mean Half-Life from Naughty Dog would adopt these qualities, so much as me being only able to gauge Naughty Dog's suitability based on the work they've produced, which when compared to Half-Life simply situates them as a talented developer who may or may not make a Half-Life game as I'd like to see it. And since that dilutes the situation down to the basics, it also makes me question why I'd chose Naughty Dog over another equally talented studio in the industry. Maybe Naughty Dog could make a true, faithful, brilliant successor to Half-Life 2, but I've no reason to believe they could do a better job of this than a number of other exceptionally talented studios. What you're left with is a fun thought experiment that, to me, just doesn't have depth to discussion. What if *insert talented developer* made *insert dead franchise*? That's what I feel this comes down to.
EDIT: As a marginally related side, spending the last few years on forums with fans discussing the death of Half-Life, I'm honestly surprised how many people fixate on "continuing/finishing the story" and the series narrative as their first point of reference for why they're disappointed Valve hasn't produced a Half-Life 3 yet.
