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This paper says that $FinVect_k$ is collectively complete for traced symmetric monoidal categories, in the sense that given distinct arrows in the free traced SMC (over some generating monoidal signature) there exists a strong functor (from the free traced SMC) into $FinVect_k$ distinguishing them.

Is there an analogous result for Cartesian categories? It would seem intuitive to me that $Set$ might be collectively complete for Cartesian categories in the sense above -- but I can't find a reference anywhere. Is this true, and is there somewhere I could find a proof?

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If I understand what you are asking, the answer is yes. Indeed:

Proposition. Let $\mathcal{C}$ be a locally small cartesian monoidal category and let $f_0, f_1 : X \to Y$ be a parallel pair of morphisms in $\mathcal{C}$. Then there is a cartesian monoidal functor $F : \mathcal{C} \to \textbf{Set}$ such that $F f_0 = F f_1$ implies $f_0 = f_1$.

Proof. Take $F = \mathcal{C} (X, -)$ and consider $\textrm{id}_X \in F (X)$. ◼

(This is basically a small part of the Yoneda lemma.)

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