It seems to me that the following must be true, and has been used in Lurie's HA, and in my question. The precise statement is:
Let $p:K \rightarrow C$ be a small diagram in $C$. Here $C$ is an $\infty$-category ,$K$ an arbitrary simplicial set. We follow HTT, 1.2.13.5, to say that $p(\infty)$ is the cone point of limit $\bar{p}$.
Claim: In $hC$, $p(\infty)$ is a $H$ -enriched limit of $hp:hK \rightarrow hC$. $H$ denotes the homotopy category of spaces.
I could not spell this out this rigorously.
The claim is that a limit in an infinity category indices an $H$-enriched limit in the homotopy category, right? In that case, this is very far from true. It would imply the same claim about ordinary limits, since an isomorphism in $H$ induces an isomorphism of sets after applying $\pi_0$. And then counterexamples are everywhere.
Generally $hC$ doesn’t have (co)limits are all, and they certainly don’t agree with the (co)limits in $C$. Pushouts are a good place to start for intuition. If $C$ is the $\infty$-category of spaces then $S^2$ is the pushout of two points along the map $S^1\to *$. But in the homotopy category, the same pushout does exist-it’s just a point.