Integration of speech into healthcare has intensified privacy concerns due to its potential as a non-invasive biomarker containing individual biometric information. In response, speaker anonymization aims to conceal personally identifiable information while retaining crucial linguistic content. However, the application of anonymization techniques to pathological speech, a critical area where privacy is especially vital, has not been extensively examined. This study investigates anonymization's impact on pathological speech across over 2,700 speakers from multiple German institutions, focusing on privacy, pathological utility, and demographic fairness. We explore both deep-learning-based and signal processing-based anonymization methods. We document substantial privacy improvements across disorders-evidenced by equal error rate increases up to 1933%, with minimal overall impact on utility. Specific disorders such as Dysarthria, Dysphonia, and Cleft Lip and Palate experience minimal utility changes, while Dysglossia shows slight improvements. Our findings underscore that the impact of anonymization varies substantially across different disorders. This necessitates disorder-specific anonymization strategies to optimally balance privacy with diagnostic utility. Additionally, our fairness analysis reveals consistent anonymization effects across most of the demographics. This study demonstrates the effectiveness of anonymization in pathological speech for enhancing privacy, while also highlighting the importance of customized and disorder-specific approaches to account for inversion attacks.