@CharlesRezk So it is certainly true that "$C_{/X}$ sifted" implies "$F\colon \mathrm{Fun}(C,\mathcal{S})\to \mathcal{S}$ preserves finite products". I don't know about the converse. I took the above reasoning from SAG 20.4.2.9. which treats the variation of your statement where we consider all finite limits, rather than just finite products.
In SAG 20.4.2.9 Lurie shows that if $F\colon \mathrm{Fun}(C,\mathcal{S})\to \mathcal{S}$ preserves finite limits, then for every finite simplicial set $K$ and every functor $p: K \to \mathrm{Fun}(C,S)$, the $\infty$-category $\mathrm{Fun}(C,S)_{/p}$ is $(-1)$-connected, i.e. non-empty. We want to show that for any functor $p: \{0,1\} \to \mathrm{Fun}(C,S)$ the category $\mathrm{Fun}(C,S)_{/p}$ is $\infty$-connected, i.e. contractible, and Lurie's method does not apply.
