You can definitely characterize groupoids as presheaves on $Fin_+$ preserving some colimtis (i.e. sending some colimits in $Fin_+$ to limits in Set). In fact Groupoids are the presheaf on $Fin_+$ that preserve the colimits comming from $\Delta$.

However, the Category $Fin_+$ has much more colimits than $\Delta$, in fact it all non-empty finite colimits. Groupoids do not preserves most of these colimits.

For example, in $Fin_+$ you have ${1} \coprod {1} \simeq {2}$ , but givena a groupoid $G$ $G(\{1\})$ is the set of objects, and $G(\{2\})$ is the set of arrow, so the isomorphisms $G(\{2\}) \simeq G(\{1\}) \times G(\{1\})$ means that your groupoid has a unique arrow between any two objects.

In fact, I claim that the colimits in $Fin_+$ that exists are exactly the finite non-empty colimits, and the category of presheaf preserving them is equivalent to the category of sets. In terms of the usual description of groupoids as presheaf on $Fin_+$, these corresponds to anti-discrete groupoids.

You can definitely characterize groupoids as presheaves on $Fin_+$ preserving some colimtis (i.e. sending some colimits in $Fin_+$ to limits in Set). In fact Groupoids are the presheaf on $Fin_+$ that preserve the colimits comming from $\Delta$.

However, the Category $Fin_+$ has much more colimits than $\Delta$, in fact it all non-empty finite colimits. Groupoids do not preserves most of these colimits.

For example, in $Fin_+$ you have $\{1\} \coprod \{1\} \simeq \{1,2\}$ , but given a a groupoid $G$, $G(\{1\})$ is the set of objects, and $G(\{1,2\})$ is the set of arrow, so the isomorphisms $G(\{1,2\}) \simeq G(\{1\}) \times G(\{1\})$ means that your groupoid has a unique arrow between any two objects.

In fact, I claim that the colimits in $Fin_+$ that exists are exactly the finite non-empty colimits, and the category of presheaf preserving them is equivalent to the category of sets. In terms of the usual description of groupoids as presheaf on $Fin_+$, these corresponds to anti-discrete groupoids.