This study explores a system of singular nonlinear higher-order katugampola fractional differential equations (KFDEs) with nonlocal coupled Riemann-Stieltjes integral boundary conditions. First, the addressed KFDEs are reformulated as the equivalent integral equations through explicit construction of some appropriate Green's functions. Second, through synergistic application of Schauder's and Guo-Krasnoselskii's fixed-point theorems, some explicit parameter interval-dependent existence criteria are established for at least one or two positive solutions to the considered KFDEs with the help of the properties of Green's functions. Finally, some concrete examples are provided to validate the effectiveness of the main theoretical results.