If we can agree that it is natural to use $\Delta$ to encode categories and their homotopical generalisations, then I don't think it is farfetched to view $\Gamma$ as the analogous gadget to encode commutative monoids and *their* homotopical generalisations.  
So let's think a little about $\Delta$ first. The category $\Delta$ has objects in bijection to $\mathbb{N}$, so that a presheaf $C$ on $\Delta$ is a graded set; for every $n \in \mathbb{N}$ we want to think of $C(n)$ as a set of chains of composable morphisms of length $n$. The morphisms in $\Delta$ are supposed to encode things like composition and associativity; so for example there should be some morphism $[1] \to [2]$ which gives composition of pairs of morphisms. But how do we know that we have all necessary morphisms in $\Delta$? For every $n \in \mathbb{N}$ denote by $\Delta_n$ the free category generated by a graph consisting of a chain of $n$ directed arrows. For a small category $\mathscr{C}$ the set of chains of morphisms of length $n$ is canonically isomorphic to $\mathbf{Cat}(\Delta_n, \mathscr{C})$. We could now hope that there is a functor $\Delta \to \mathbf{Cat}, \; [n] \mapsto \Delta_n$ which induces the fully faithful functor $\mathbf{Cat} \hookrightarrow \widehat{\Delta}, \;  \mathscr{C} \mapsto ([n] \mapsto \mathbf{Cat}(\Delta_n,\mathscr{C}))$. Indeed, if we choose this functor to be fully faithful, which completely determines the structure of $\Delta$, then we obtain such a functor $\mathbf{Cat} \hookrightarrow \widehat{\Delta}$, the classical nerve functor.   
If we play the same game with monoids we obtain the $\Gamma$ category. If we view any commutative monoid as a category, then we see that we might again suppose that $\Gamma$ has objects in bijection to $\mathbb{N}$; for any number $n \in \mathbb{N}$ we call the corresponding object $(n)$. Now a commutative monoid has exactly one object, so without knowing anything else, we may already assume that $(0)$ gets mapped to $\{*\}$. Like for $\Delta$, let's try to view $\Gamma$ as a subcategory of $\mathbf{AbMon}$, the category of commutative monoids, in order to again obtain a nerve functor. The objects have to be $\mathbb{N} \oplus \underbrace{\cdots}_{n \times} \oplus \mathbb{N}$ for all $n \in \mathbb{N}$. The morphisms of $\Gamma$ are then completely determined by where the generators of $\mathbb{N} \oplus \underbrace{\cdots}_{n \times} \oplus \mathbb{N}$ are sent. To get an appropriate subcategory of $\mathbf{AbMon}$ we will only consider those morphisms which send generators to elements, which themselves are sums of distinct generators (this will be important below). _If we label and keep track of only the generators, we rediscover Segal's original description of $\Gamma$._ The induced nerve functor is again fully faithful, and it is now straightforward how to view any commutative monoid as a $\Gamma$-set. Furthermore, we note that there is a unique morphism $(n) \to (0)$ for all $n \in \mathbb{N}$, so that any presheaf on $\Gamma$ taking $(0)$ to $\{ * \}$ factors through the category of pointed sets $\mathbf{Set}_*$, and we might as well consider contravariant functors $\Gamma^{\mathrm{op}} \to \mathbf{Set}_*$.   
Finally, let us show that there is a canonical functor $\Delta \to \Gamma$, so that every $\Gamma$-set has an underlying simplicial set. This is simple: by again viewing any commutative monoid as a category, we simply send the generators of $\Delta_n$ to the generators of $\mathbb{N} \oplus \underbrace{\cdots}_{n \times} \oplus \mathbb{N}$ for every $n \in \mathbb{N}$. In this last step we are implicitly using that we have labelled the generators of $\mathbb{N} \oplus \underbrace{\cdots}_{n \times} \oplus \mathbb{N}$, but this is not a problem: Let us denote the nerve functor $\mathbf{Cat} \hookrightarrow \widehat{\Delta}$ by $N$, and the nerve functor $\mathbf{AbMon} \hookrightarrow \mathbf{Cat}(\Gamma^{\mathrm{op}}, \mathbf{Set}_*)$ by $N_\Gamma$. Then for any monoid $M$, any functor $\Delta \to \Gamma$ induced by any labelling will take $N_\Gamma(M)$ to $N(M)$ (viewing $\mathbf{AbMon}$ as a subcategory of $\mathbf{Cat}$). I don't, however, know whether arbitrary presheaves $\Gamma^{\mathrm{op}} \to \mathbf{Set}_*$ get mapped to the same simplicial set under the different functors induced by different labellings.