Finite limits and the topology
For the finite-limit assertion, use the description of limits via cofree games in Hora, Appendix B.2–B.3. A finite nonempty product embeds into the cofree game on the finite product of the carriers. Each projection preserves the length of a longest play, so the product has bounded height. On a finite set of labels, the bounded-height part of the cofree game is finite, by induction on height. Equalizers are subgames, hence remain finite. This gives finite nonempty limits. Finite colimits are finite because the forgetful functor creates colimits.
In particular, if $B\to A$ is the surjection associated with a finite covering family, its kernel pair $B\times_A B$ is finite. This follows from the finite-limit assertion, not from assuming that its underlying set is the set-theoretic kernel pair.
The Page proof of subcanonicity now uses the finite-game site presentation instead. That presentation is obtained independently from exact completion, Giraud's theorem and the Comparison Lemma; it does not assume subcanonicity of $J$ or any prior property of the nerve. The representables and the nerves of all games are consequently sheaves for $J$.