Large language models (LLMs) show promise for supporting systematic reviews (SR), even complex tasks such as qualitative synthesis (QS). However, applying them to a stage that is unevenly reported and variably conducted carries important risks: misuse can amplify existing weaknesses and erode confidence in the SR findings. To examine the challenges of using LLMs for QS, we conducted a collaborative autoethnography involving two trials. We evaluated each trial for methodological rigor and practical usefulness, and interpreted the results through a technical lens informed by how LLMs are built and their current limitations.