That's what "Assignments" and InCopy are for...
Our team don't use InCopy.... the fact "Assignments" are never mentioned wrt InDesign in isolation led me to believe they weren't what we were looking for? (Can only be accessed in the context of working with InCopy?)
Our problem is specifically around multiple designers working on InDesign files and assets simultaneously.
If you have to do this, then use InCopy, as you have it. What is hindering you? One InDesign file can only be opened on a single computer by a single user. But if you have asignments you can check out them from different users at the same time.
You could also split up your INDD in several documents and use a book file.