Clustering is a fundamental problem in machine learning and operations research. Therefore, given the fact that fairness considerations have become of paramount importance in algorithm design, fairness in clustering has received significant attention from the research community. The literature in fair clustering has resulted in a collection of interesting fairness notions and elaborate algorithms. In this paper, we take a critical view of fair clustering, identifying a collection of ignored issues such as the lack of a clear utility characterization and the difficulty in accounting for the downstream effects of a fair clustering algorithm in machine learning settings. In some cases, we demonstrate examples where the application of a fair clustering algorithm can have significant negative impacts on social welfare. We end by identifying a collection of steps that would lead towards more impactful research in fair clustering.