In this study, we conducted an in-depth examination of the utility analysis of the differentially private mechanism (DPM). The authors of DPM have already established the probability of a good split being selected and of DPM halting. In this study, we expanded the analysis of the stopping criterion and provided an interpretation of these guarantees in the context of realistic input distributions. Our findings revealed constraints on the minimum cluster size and the metric weight for the scoring function. Furthermore, we introduced an interpretation of the utility of DPM through the lens of the clustering metric, the silhouette score. Our findings indicate that even when an optimal DPM-based split is employed, the silhouette score of the resulting clustering may still decline. This observation calls into question the suitability of the silhouette score as a clustering metric. Finally, we examined the potential of the underlying concept of DPM by linking it to a more theoretical view, that of $(ξ, ρ)$-separability. This extensive analysis of the theoretical guarantees of DPM allows a better understanding of its behaviour for arbitrary inputs. From these guarantees, we can analyse the impact of different hyperparameters and different input data sets, thereby promoting the application of DPM in practice for unknown settings and data sets.