We consider the limiting behavior of the count of subgraphs isomorphic to a graph $G$ with $m\geq 0$ fixed endpoints (or roots) in the random-connection model, as the intensity $λ$ of the underlying Poisson point process tends to infinity. When connection probabilities are of order $λ^{-α}$ we identify a phase transition phenomenon depending on a critical decay rate $α^\ast_m (G)>0$ such that normal approximation for subgraph counts holds when $α\in (0,α^\ast_m (G) )$, and a Poisson limit result holds if $α= α^\ast_m (G)$. Our approach relies on cumulant growth rates derived by the convex analysis of planar diagrams that enumerate the partitions involved in cumulant identities. As a result, by the cumulant method we obtain normal approximation results with convergence rates in the Kolmogorov distance, and a Poisson limit theorem, for subgraph counts.