GitHub seem to have solved both the lack of builds issue and the challenge of deploying multiple branches (their bot can deploy a branch to staging with a single command) - which just leaves refactoring. I imagine they deal with refactoring by broadcasting a message to the team saying "I'm working on refactoring area X, try not to touch that code until I'm done if you don't want to deal with a painful merge".

I intend no ill will with this response, but it is worth pointing out that "the first two" "bigger issues" have nothing to do with your revision control system.

Lack of builds is something about which you and your CI server need to have a chat, and refactoring is something about which you and your people need to have numerous chats before, during, and afterwards.