Magnetic dilution of a well-established Kitaev candidate system is realized in the substitutional Ru$_{1-x}$Rh$_x$Cl$_3$ series ($x = 0.02-0.6$). Optimized syntheses protocols yield uniformly-doped single crystals and polycrystalline powders that are isostructural to the parental $\alpha$-RuCl$_3$ as per X-ray diffraction. The Rh content $x$ is accurately determined by the quantitative energy-dispersive X-ray spectroscopy technique with standards. We determine the magnetic phase diagram of Ru$_{1-x}$Rh$_x$Cl$_3$ for in-plane magnetic fields from magnetization and specific-heat measurements as a function of $x$ and stacking periodicity, and identify the suppression of the magnetic order at $x \approx 0.2$ towards a disordered phase, which does not show any clear signature of freezing into a spin glass. Comparing with previous studies on the substitution series Ru$_{1-x}$Ir$_x$Cl$_3$, we propose that chemical pressure would contribute to the suppression of magnetic order especially in Ru$_{1-x}$Ir$_x$Cl$_3$ and that the zigzag magnetic ground state appears to be relatively robust with respect to the dilution of the Kitaev--$\Gamma$--Heisenberg magnetic lattice. We also discovered a slight dependence of the magnetic properties on thermal cycling, which would be due to an incomplete structural transition.
