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The hierarchical structure inherent in many real-world datasets makes the modeling of such hierarchies a crucial objective in both unsupervised and supervised machine learning. While recent advancements have introduced deep architectures specifically designed for hierarchical clustering, we adopt a critical perspective on this line of research. Our findings reveal that these methods face significant limitations in scalability and performance when applied to realistic datasets.~Given these findings, we present an alternative approach and introduce a lightweight method that builds on pre-trained non-hierarchical clustering models. Remarkably, our approach outperforms specialized deep models for hierarchical clustering, and it is broadly applicable to any pre-trained clustering model that outputs logits, without requiring any fine-tuning. To highlight the generality of our approach, we extend its application to a supervised setting, demonstrating its ability to recover meaningful hierarchies from a pre-trained ImageNet classifier. Our results establish a practical and effective alternative to existing deep hierarchical clustering methods, with significant advantages in efficiency, scalability and performance.