Eventually got it to work, despite non-existent introduction/use tutorial instructions, bad user interface (that is, for me as the WordPress site administrator -- works fine for the site visitor).
Screenshots do not a tutorial make! Some basic instructions to create a file download button:
1. Use File Manager > Add new file to upload a file, which it calls a "Download Package". Ignore the main part of the window, the text editor screen. It does nothing! Don't select an icon -- it screwed up the button when I tried it.
2. Go to File Manager > Manage to find out what ID it's given your file.
3. When you're editing a page or post, click the new TinyMCE button, "Download Manager: Insert Package or Category." Use the "Insert Shortcode" tab, and click the first "Insert into Post". Make sure the id= part matches the ID from step 2. (No, it doesn't let you choose which file to make downloadable!)
And yes, I wasted a couple of hours figuring this out. Please Shoan, let your users accomplish the simple tasks simply!