Recent breakthroughs in Automatic Speech Recognition (ASR) have enabled fully automated Alzheimer's Disease (AD) detection using ASR transcripts. Nonetheless, the impact of ASR errors on AD detection remains poorly understood. This paper fills the gap. We conduct a comprehensive study on AD detection using transcripts from various ASR models and their synthesized speech on the ADReSS dataset. Experimental results reveal that certain ASR transcripts (ASR-synthesized speech) outperform manual transcripts (manual-synthesized speech) in detection accuracy, suggesting that ASR errors may provide valuable cues for improving AD detection. Additionally, we propose a cross-attention-based interpretability model that not only identifies these cues but also achieves superior or comparable performance to the baseline. Furthermore, we utilize this model to unveil AD-related patterns within pre-trained embeddings. Our study offers novel insights into the potential of ASR models for AD detection.