Hi - part of the issue is that Work On It changes a task or issue from a Work Request to Work Item. This is legacy functionality from the old My Work area where Work List was separate from Work Requests.
A fix that worked at my old Agency was including Can Start in the right side area. Obviously, they can't filter out the Can Start equals False in the work list, but at least they can easily see in the details area if they can start on the task or not. (To be 100% honest, a lot of my business units ended up using My Work dashboards we created instead of using Home by the time I left)
Also, here is an idea in the Innovation Lab that says if you use the filter Ready To Start, it filters the Team Requests as well. Might not help your team since it seems you pick up work before it is ready to start, but thought I'd still share: https://one.workfront.com/s/idea/0870z000000PS6ZAAW/detail