Despite numerous efforts to mitigate their biases, ML systems continue to harm already-marginalized people. While predominant ML approaches assume bias can be removed and fair models can be created, we show that these are not always possible, nor desirable, goals. We reframe the problem of ML bias by creating models to identify biased language, drawing attention to a dataset's biases rather than trying to remove them. Then, through a workshop, we evaluated the models for a specific use case: workflows of information and heritage professionals. Our findings demonstrate the limitations of ML for identifying bias due to its contextual nature, the way in which approaches to mitigating it can simultaneously privilege and oppress different communities, and its inevitability. We demonstrate the need to expand ML approaches to bias and fairness, providing a mixed-methods approach to investigating the feasibility of removing bias or achieving fairness in a given ML use case.