Its essential to regularly update what tasks are done and what tasks are being worked on by whom, but if you use some other system to handle those updates and keep people aware of the updates (e.g., visual artifacts updated as items are completed and pulled for work, as in kanban), then you don't need to do that in the standup meeting.

Identifying issues encountered (both to share knowledge of how issues were resolved and to gather information on addressing barriers that haven't been resolved) is a pretty standard element of standup meetings even when they include progress updates, and certainly a good thing to keep them for if you've moved the status updates to another channel.