$\mathfrak C[S]$ is defined to be the colimit, among simplicial categories, of copies of $\mathfrak C[\Delta^n]$ indexed by the category of maps from simplices into $S$. This implies in particular that it’s a simplicial category, simply because these are closed under colimits. The objects of $\mathfrak C[S]$ are just the $0$-simplices of $S$. Other than by direct inspection, one way to check this quickly is to observe that the functor from simplicial set to sets giving the set of $0$-simplices factors through $\mathfrak C$, simply because it does so when restricted to the objects $\Delta^n$.
I suggest to use profunctors induced by (or simply instead of) partial functors.
First of all, a profunctor $H:A^{op}\times B\to {\rm Set}$ is said to be functorial if $H(a,-):B\to {\rm Set}$ is a representable functor for each object $a\in A$. This is equivalent to saying that $H\simeq \hom_B(F-,\, -)\,=:F_*$ for a functor $F:A\to B$.
And actually the profunctor of the colimit functor can be defined for any category, even if it lacks some colimits of the given shape: namely, consider
$$H:(C^I)^{op}\times C\to {\rm Set}\quad (\underset{I\to C}D,\,a)\mapsto
\{D\to a\text{ cocones in }C\}\,,$$
where a $D\to a$ cocone can be thought of as a natural transformation $D\to a$ to the constant functor.
Note that a diagram $D\in C^I$ has a colimit, by the very definition, if and only if $H(D,-)$ is representable.
Secondly, not only partial functors but also all categorical relations (that is, subcategories of $A\times B$), moreover any span of categories $A\overset{P}\leftarrow R\overset{Q}\to B$ do determine a profunctor in a natural way, namely,
$$P^*\otimes Q_*:A\not\to R\not\to B$$
where $P^*=\hom_A(-,\,P-);\ \ Q_*=\hom_B(Q-,\,-)$ and $\otimes$ means composition (tensor product) of profunctors, which is a colimit/coend construction that yields a quotient of $\displaystyle\bigsqcup_{r\in R}P^*(a,r)\times Q_*(r,b)$.
I guess, the $2$-category you are defining would naturally embed into the bicategory of profunctors this way.
Best Answer
For a counterexample, consider the span of categories $\def\B{{\sf B}}\def\Z{{\bf Z}}1←\B\Z=S^1→1$, where 1 denotes the terminal groupoid and $\B$ denotes the delooping functor.
The colimit of this span in the (2,1)-category Cat is the terminal groupoid 1.
The colimit of this span in the (∞,1)-category (∞,1)Cat is the (∞,1)-groupoid $S^2$.