Have no fear. After poking around a bit, it turns out Teams is almost as flexible as SharePoint.
- In Teams, click on the Files tab for your channel.
- Click 'Open in SharePoint'.
- Edit the view, or create a new one, and add the desired metadata.
- Copy the web address (something like, https://xxx.sharepoint.com/sites/ComeTogether/Shared%20Documents/Forms/AllItems.aspx)
- Back in Teams, add a new tab by clicking +.
- Search for 'web' and add the Website type tab.
- Fill out the Tab name and URL.
- Golden!
Related, someone asked me, "Wouldn't it be cool if there were a place to drop files that all channels could see, like at the root of the Shared Documents library that already has folders for each of my channels?
Why, yes. Yes, it would.
- Follow the steps above.
- In the SharePoint view you create, set the Filter to Content Type is not equal to Folder.
- Celebrate!