The low-temperature unitary Bose gas is a fundamental paradigm in few-body and many-body physics, attracting wide theoretical and experimental interest. Here, we present experiments performed with unitary ¹³³Cs and ⁷Li atoms in two different setups, which enable quantitative comparison of the three-body recombination rate in the low-temperature domain. We develop a theoretical model that describes the dynamic competition between two-body evaporation and three-body recombination in a harmonically trapped unitary atomic gas above the condensation temperature. We identify a universal "magic" trap depth where, within some parameter range, evaporative cooling is balanced by recombination heating and the gas temperature stays constant. Our model is developed for the usual three-dimensional evaporation regime as well as the two-dimensional evaporation case, and it fully supports our experimental findings. Combined ¹³³Cs and ⁷Li experimental data allow investigations of loss dynamics over 2 orders of magnitude in temperature and 4 orders of magnitude in three-body loss rate. We confirm the 1/T² temperature universality law. In particular, we measure, for the first time, the Efimov inelasticity parameter η_*=0.098(7) for the 47.8-G d-wave Feshbach resonance in ¹³³Cs. Our result supports the universal loss dynamics of trapped unitary Bose gases up to a single parameter η_*.