We have conducted a revised analysis of the first-order phase transition that is associated with symmetry breaking in a classically scale-invariant model that has been extended with a new $SU(2)$ gauge group. By incorporating recent developments in the understanding of supercooled phase transitions, we were able to calculate all of its features and significantly limit the parameter space. We were also able to predict the gravitational wave spectra generated during this phase transition and found that this model is well-testable with LISA. Additionally, we have made predictions regarding the relic dark matter abundance. Our predictions are consistent with observations but only within a narrow part of the parameter space. We have placed significant constraints on the supercool dark matter scenario by improving the description of percolation and reheating after the phase transition, as well as including the running of couplings. Finally, we have also analyzed the renormalization-scale dependence of our results.