I've not found that Dropbox is a problem, it's definitely more stable than any other shared files solution I've worked with.
It sounds like you're asking if you can use the CC Files folder for the InDesign/InCopy workflow. In some ways you can. And no, you don't need to upgrade to Teams to do so.
In your CC Files folder, create a project folder and put the InDesign file and the exported InCopy stories (ICML files) inside it. I'd suggest you "hide" the ICML files inside a subfolder, named something like "stories" or "text" to help your editors find what they need. Then share that folder with your editor. (Right-click on the project folder in your CC Files folder on your hard drive and choose "Collaborate ..."). You'll need to know the email address they use to log on to their CC account, this is the email address you enter in the Collaborate dialog box. Be sure you choose "Can Edit" for their permissions.
Your InCopy-using colleagues will need to "accept" the folder; they'll get notified in their CC app that you're requesting to share it with them. After they say okay, the contents of your folder will be synced to/downloaded to their own CC Files folder on their own hard drive.
Up to this point, it's exactly how Dropbox works.
With CC Files (and Dropbox), your editors will be able to open the local InDesign layout file in InCopy (use File > Open from InCopy) and check out stories and edit them, or they could open an individual ICML file and edit it. They can save their changes and close the files, checking them in. They need to keep all files in their local CC Files folder without renaming or moving them.
What CC Files is incapable of doing is notifying you or others that someone has checked out a file. You won't see the "in use" icon. So, it's a little dangerous. Dropbox, on the other hand, does sync the checked-in/checked-out states of story files.
As long as you're careful not to edit the same file at the same time (by checking w/each other first by phone or email, for example), you could use CC files in this way.
Assignments in either case are unnecessary. In InDesign, just choose Edit > InCopy > Export > All Stories, and as I said, choose a subfolder at the same level as the ID file to export the stories to. The InCopy user/s open the full layout and check out stories within. The only way InCopy will open an InDesign layout though is by either using InCopy's own File > Open menu (and then selecting an INDD file) or the user could associate the ".indd" file extension to the InCopy app in their operating system, if they know how to do that.
AM