This is just an expended version of the comment. The answer to the question as asked is no.

The problem is that for any ($\infty$-)category $J$ the category $D_J$ of functors $J \to Cat_\infty$ that are levelwise coproduct of the $\Delta[n]$ is a $1$-category. So if the Identity of $Cat_\infty$ could be factored through $Colim: D_J \to Cat_\infty$ then it would factor through some $1$-category, and hence through the homotopy category of $Cat_\infty$.

This is of course impossible as that would make $Cat_\infty$ equivalent to a $1$-category, or more explicitly, show that any invertivle natural transformation $\alpha:F \to F$ is isomorphic to the identity.

What you can do if you want such a functorial, is to replace "coproducts" by $\infty$-groupoid indexed colimit. Then we can simply observe that because complete Segal spaces are a full subcategory of simplicial spaces that any (complete) Segal space can be written as the coend

$$ X = \int^{[n] \in \Delta} X_n \times \Delta[n]$$

and then follow the discussion as in the OP, but this time realize $X$ as a colimits of things of the form $S \times \Delta[n]$ where $S$ is a space (i.e. an $\infty$-groupoid) instead of a set.

I should say this is very much in the spirit of the discussion here: If you don't want to break the equivalence principles, ($\infty$-)categories are not structure on sets, they are structure on ($\infty$-)groupoids.

This is just an expended version of the comment. The answer to the question as asked is no.

The problem is that for any ($\infty$-)category $J$ the category $D_J$ of functors $J \to Cat_\infty$ that are levelwise coproduct of the $\Delta[n]$ is a $1$-category. So if the Identity of $Cat_\infty$ could be factored through $Colim: D_J \to Cat_\infty$ then it would factor through some $1$-category, and hence through the homotopy category of $Cat_\infty$.

This is of course impossible as that would make $Cat_\infty$ equivalent to a $1$-category, or more explicitly, show that any invertivle natural transformation $\alpha:F \to F$ is isomorphic to the identity.

What you can do however if you want such a functorial expressions of categories as colimits of their simplicies, is to replace "coproducts" by $\infty$-groupoid indexed colimit. Then we can simply observe that because complete Segal spaces are a full subcategory of simplicial spaces that any (complete) Segal space can be written as the coend

$$ X = \int^{[n] \in \Delta} X_n \times \Delta[n]$$

and then follow the discussion as in the OP, but this time realize $X$ as a colimits of things of the form $S \times \Delta[n]$ where $S$ is a space (i.e. an $\infty$-groupoid) instead of a set.

I should say this is very much in the spirit of the discussion here: If you don't want to break the equivalence principles, ($\infty$-)categories are not structure on sets, they are structure on ($\infty$-)groupoids. So I've taken the simplex construction in the beginning and replaced it with the $\infty$-groupoid of $n$-simplex.