In the recent literature, the conception has emerged that supercell tornado potential may mostly depend on the strength of the low-level updraft, with more than sufficient subtornadic vertical vorticity being assumed to be present in the outflow. In this study, we use highly idealized simulations with heat sinks and sources to conduct controlled experiments, changing the cold pool or low-level updraft character independently. Multiple, time-dependent heat sinks are employed to produce a realistic near-ground cold pool structure. It is shown that both the cold pool and updraft strength actively contribute to the tornado potential. Furthermore, there is a sharp transition between tornadic and nontornadic cases, indicating a bifurcation between these two regimes triggered by small changes in the heat source or sink magnitude. Moreover, larger updraft strength, updraft width, and cold pool deficit do not necessarily result in a stronger maximum near-ground vertical vorticity. However, a stronger updraft or cold pool can both drastically reduce the time it takes for the first vortex to form.