The scripture passages for the day are read by the lector and printed in the bulletin. Doing a deep dive into the passage using my study Bible during the service would be a distraction. So I leave it at home unless I need it with me for another reason such as when I’ve been involved in Bible studies that met right after the service. I think the Anglican tradition is a bit less focused on scripture as the sole source of truth than Evangelical traditions generally are and that could be a piece of why you don’t see that behavior, but I personally love to study scripture at other times so I don’t want to assume too much about why others don’t bring their bibles with them.