Automatic Speech Recognition (ASR) is a critical component of any fully-automated speech-based Alzheimer's disease (AD) detection model. However, despite years of speech recognition research, little is known about the impact of ASR performance on AD detection. In this paper, we experiment with controlled amounts of artificially generated ASR errors and investigate their influence on AD detection. We find that deletion errors affect AD detection performance the most, due to their impact on the features of syntactic complexity and discourse representation in speech. We show the trend to be generalisable across two different datasets and two different speech-related tasks. As a conclusion, we propose changing the ASR optimization functions to reflect a higher penalty for deletion errors when using ASR for AD detection.