Best Of

This sounds like a combination of things: Onshape could add some more guard rail functionality, AND a best practice needs to be developed and made explicit at your company. I personally don't see value in the twigs thing, and it could add an exponential level of complexity to the version graph.

Functionality I'm interested in:

• granular permissions for branches - There's value in thinking of each branch as "owned" by a certain person, and it could be valuable to give the ability to prevent everyone but the owner of a branch for editing it. There could be a single "Branch Settings" dialogue for branch permissions which lets you specify users, and certain action limitations (like releasing).
• selecting which branches are visible to import or search - As you say, it can become cluttered when you have 6 copies of the same part in 6 branches. It would be great to have a way to mark some branches as important, and others as not. It could be as easy as the "critical" branch items always sort to the top, but they're all still there. Even this could be part of the Branch Settings menu.
• An easy way to tell who made which branch - As with all collaborative environments, it's easier to add things than clean them up because it's risky to remove clutter that you didn't make and may not understand. It's the same reason shop spaces are usually a mess. Things flow into them freely, but no one knows what's garbage. If there were an easy way for people to know who made a branch they can ask them if it can be deleted now.

But there will always be a branch hygiene element to it, which is a sociological problem, not a software problem. It's about setting and keeping SoPs for how branches are and aren't used. An easy one could be to ask people to append their name to the branch name so it's obvious who to ask about it. Another could be getting clear on the CAD strategy before beginning a project. Before any work is done the whole project team should understand how data flows through the design, and at what milestone the project should split into multiple documents. (Often a good practice is to start in 1 doc while things are changing a lot, then split out once the design is beginning to gel). No amount of Onshape functionality can replace the team's mutual understanding of the CAD plan and SoPs. It's a compound challenge involving people, process, and tools, and the solution needs to address all 3. I help with this stuff at www.theonsherpa.com.