What are the examples of “tame” minimal Kan simplicial sets having finite number of simplexes in each dimension besides simplicial point and $B(G)\approx K(G,1) $ for a finite group $G$? I believe that Alain Connes’ simplicial circle is also minimal Kan.
A tame minimal Kan complex has finite homotopy groups. The converse is also true: if homotopy groups of a minimal Kan complex are finite, then there are finitely many $n$-simplices. The proof is by induction on $n$. First, $X_0 = \pi_0(X)$ is finite. If $n > 0$, there are finitely many possible choices of boundaries for $n$-simplices by induction hypothesis. For every such boundary, the set of its fillers is either empty or isomorphic to $\pi_n(X,x_0)$ for some $x_0$. This completes the proof. This fact implies that, for every Kan complex with finite homotopy group, there is an equivalent tame minimal Kan complex.