Single-shot all-optical helicity-independent switching (AO-HIS) induced by a femtosecond laser pulse has been mainly reported in gadolinium-based rare-earth (RE)–transition-metal alloys such as GdFeCo or GdCo, but the mechanism leading to magnetization switching is a hotly debated topic. Here, we elaborate on a large number of Gd$_y$RE$_{1−x−y}$Co$_x$ (RE = Dy, Tb, Ho) alloys to tune various magnetic parameters in order to define what the criteria are for observing AO-HIS in such systems. The state diagrams show that two laser-fluence thresholds must be considered: the fluence which induces the single laser-pulse switching (F$_{switch}$) and the fluence at which the material breaks into a multidomain state (F$_{multi}$). Those two fluences are shown to behave very differently as a function of the material properties and the laser-pulse duration. The criterion to ensure the observation of AO-HIS is F$_{multi}$ > F$_{switch}$. Taking into account the parameters defining the conditions for which multidomain states are created and considering only the angular momentum transfer from the Gd sublattice to the rest of the system explains in large our experimental results. The importance of the compensation in the ferrimagnetic alloys is also discussed. We believe the defined criteria will be an important tool for designing ultrafast spintronic devices based on all-optical switching.