In this paper, we propose a training-free method for unsupervised short text clustering that relies less on careful selection of embedders than other methods. In customer-facing chatbots, companies are dealing with large amounts of user utterances that need to be clustered according to their intent. In these settings, no labeled data is typically available, and the number of clusters is not known. Recent approaches to short-text clustering in label-free settings incorporate LLM output to refine existing embeddings. While LLMs can identify similar texts effectively, the resulting similarities may not be directly represented by distances in the dense vector space, as they depend on the original embedding. We therefore propose a method for transforming LLM judgments directly into a bag-of-texts representation in which texts are initialized to be equidistant, without assuming any prior distance relationships. Our method achieves comparable or superior results to state-of-the-art methods, but without embeddings optimization or assuming prior knowledge of clusters or labels. Experiments on diverse datasets and smaller LLMs show that our method is model agnostic and can be applied to any embedder, with relatively small LLMs, and different clustering methods. We also show how our method scales to large datasets, reducing the computational cost of the LLM use. The flexibility and scalability of our method make it more aligned with real-world training-free scenarios than existing clustering methods.