Suppose that we have a functor $F : \boldsymbol{\Delta}^\bullet \to \mathsf{C}$ with domain the full subcategory of simplicial sets given by representable functors. For example, for each $\Delta^n = \hom(n,-)$ we can assign to it its baricentric subdivision $\mathsf{sd} \Delta^n \in \mathsf{sSet}$, or its geometric realization $|\Delta^n| \in \mathsf{Top}$.
By the Yoneda embedding, we have a fully faithful injective on objects functor $i: \Delta^{op} \hookrightarrow \boldsymbol{\Delta}$, hence $F$ can be thought of as a simplicial object
$$ F : \Delta^{op} \to \mathsf{C}. $$
On the other hand, if $X$ is any simplicial set, we know that it is a colimit of representables
$$ X = \mathsf{colim}_{\Delta^n \to X} \Delta^n. $$
If $\mathsf{C}$ is cocomplete, the definition
$$ \widetilde{F}X := \mathsf{colim}_{\Delta^n \to X} F\Delta^n, \tag{1}$$
makes sense and gives an extension of $F$ to a functor $\widetilde{F} : \mathsf{sSet} \to \mathsf{C}$.
In other terms, we are using that simplicial sets are the free cocompletion of $\Delta$, and so this is the universal cocontinuous extension of $F$.
If I am not mistaken, since $Fk = F\Delta^k$, using the cone leg arrows the maps
$$ Fk \to F\Delta^k \hookrightarrow \mathsf{colim}_{\Delta^n \to X} F\Delta^k= \widetilde{F}\Delta^n $$
gives a natural transformation $\eta : F\Rightarrow \widetilde{F}i$. So, assuming the former is correct, my question is:
Is $(\widetilde{F},\eta)$ a left Kan extension of $F$ along $i$?
I would also be interested in knowing what happens when we consider right Kan extensions, if these coincide and if not, what other interesting extension constructions can be made.
The fact that every functor $F$ like yours, with cocomplete codomain, admits a (essentially unique ) extension to $sSet$ amounts to the universal property of the free cocompletion, yes; and yes, the extension (has a right adjoint, called the $F$-nerve) and is a left Kan extension, along the Yoneda embedding $y : \Delta \to {\sf sSet}$.
There is plenty of places where this is proved, but I can't help from the usual self-promotion: Theorem 3.1.1 here.
As for right extensions, that's another story: the opposite of the category of presheaves on $\Delta^{op}$, i.e. the category $[\Delta, {\sf Set}]^{op}$, exhibits the universal property of the free completion of $\Delta$, and the contravariant Yoneda embedding $y^\sharp : \Delta^{op}\to [\Delta, {\sf Set}]$ yield a continuous extension for every $G$ with complete domain.
Usually, even assuming $\sf C$ bicomplete, it is not the case that $\text{Lan}_y F \cong \text{Ran}_{y^\sharp} F$.