You have made an error in interpreting those papers. The product family is *not* $\{\otimes_{i=1}^n P_\theta\}_{\theta \in \Theta}$, it is $\{P_{\theta_1} \otimes \dots \otimes P_{\theta_n}\}_{(\theta_1, \dots, \theta_n) \in \Theta^n}$.

The statistic $S$ is minimal sufficient for the latter family, whereas $R$ is minimal sufficient for the (much smaller) former family.