@TimCampion Thanks for the link (and for asking the question on MO)! Rune's and Dylan's answer is the same as I came up with, when I first started thinking about the problem, and constitutes another reason why I tried to link Lurie's construction with coCartesian fibration.
At this point, I reckon that Lurie's construction might really best be thought of as a neat way of constructing a simplicial set (which in general is not a quasi-category) which corepresents a functor $K \to \mathcal{C}$ together with cone on all the restrictions to $K_i$, which hang together nicely.
Obstruction theory for sections of a map f:X-->Y can be worked out using attempts at lifting sections along the Postnikov filtration of f, or lifting sections of the pullback of f to a filtration of Y by a choice of skeleta (which is then independent of the choice). There's a place where the equivalence of the two approaches is proven?
