We report a continuous-wave operation of Tm:YAG and Tm:LuAG lasers pumped with a low-cost, multimode AlGaAs laser diode. First, the lifetime and the absorbance behavior of 5 mm, 6 % Tm$^{3+}$-doped YAG and LuAG crystals were thoroughly investigated. A low-cost multimode 3W laser diode at 781 nm was then used as a pump source for the Tm$^{3+}$-doped laser systems. Using three different output couplers, up to 636 mW of output power was obtained from Tm:YAG laser, with a slope efficiency of 29 % at 2017 nm. The maximum output power was 637 mW in the Tm:LuAG laser, with a slope efficiency of 28 % at 2023 nm. The lasing performances showed a decreasing slope efficiency with an increasing level of output coupling, which leads to a high upconversion. Furthermore, using a birefringent filter in the resonators, the laser outputs were tuned from 1942 to 2086 nm in the Tm:YAG resonator and from 1931 to 2107 nm in the Tm:LuAG case.