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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.