Disinformation poses a growing challenge to content moderation on online platforms, motivating the development of automated approaches for information verification, such as Large Language Model (LLM)-driven fact-checkers. In this position paper, we discuss challenges for responsible LLM-driven fact-checking from a user-centered perspective. We state that while LLMs hold considerable potential to help users assess information online, the use of LLMs requires a trustworthiness-by-design approach and meaningful human oversight. Consequently, the use of LLMs in human-centered content moderation should augment rather than replace human judgment.