That sounds exactly like a system I set up with Jenkins a while back. It would deploy to a test server and run the tests from any branch. If you pushed to the deploy branch, it would deploy to the real server assuming all the tests passed.

The actual deploying and testing was all done with Fabric, but the idea was the same.

I think that's a pretty good case for using a CI system like Jenkins, especially because it's really easy to set up. It didn't support Git by default, but there was a plugin that was both easy to install and fairly comprehensive.