Exact-completion criterion and the finite-game site
Let $\mathcal C$ be a regular, infinitary extensive category, and let $I:\mathcal C\to\mathcal E$ be its exact/regular completion. Suppose there is a set of regular generators: every object of $\mathcal C$ is a regular quotient of a coproduct of objects from this set.
By Lack, Proposition 3.2, $I$ is fully faithful and regular and $\mathcal E$ is exact. Every object of $\mathcal E$ is a regular quotient of some $I(X)$. By Vitale, p.237, Lemma (3), $\mathcal E$ is infinitary extensive and $I$ preserves coproducts. Composing regular quotient maps now shows that the images of the given generators regularly generate $\mathcal E$. Giraud's theorem makes $\mathcal E$ a Grothendieck topos; compare Worrell, Theorem 5.5.1 and the proof of Corollary 5.5.7.
For Games, the coproduct of the finite subgames generated by individual positions maps surjectively onto any game. These are regular quotients by the regularity proposition on Games. This verifies the generating hypothesis without using the nerve.
Identification of the topology
Use the images of finite games as a full generating subcategory of $\mathcal E$. The generating-subcategory form of the Comparison Lemma, Caramello §3.2 presents $\mathcal E$ as sheaves on this subcategory with its induced canonical topology.
For a family $A_i\to A$ of finite games, the corresponding family in $\mathcal E$ is jointly epimorphic exactly when $\coprod_i A_i\to A$ is surjective. To check this, $I$ preserves coproducts and regular epimorphisms. It also reflects regular epimorphisms: factor a map into a regular epi and a mono in $\mathcal C$; if its image under $I$ is a regular epi, the image mono is invertible, hence so is the original mono. Thus the induced covers are precisely the jointly surjective families. Such a family onto a finite game has a finite subcover, including the empty subcover of the empty game. This is the stated topology $J$.
Under the comparison equivalence, $I(X)$ is sent to the sheaf $A\mapsto\mathcal E(I(A),I(X))$. Full faithfulness identifies this with $A\mapsto\mathbf{Games}(A,X)$. This proves the site presentation and the identification of the nerve before any of its exact-completion properties are used.