Thanks for your post- you're correct; currently SQL Source Control just links to a working folder.
The commands we run to do a commit etc. are stored in an XML file - you can click "manage config files" on the link page, and create a copy of the git.xml to edit for your own purposes... so it *may* be possible to create, say, a batch file that does a commit then a push as two steps, and call your batch file in git.xml. I don't know of anyone who's tried this; so there's no guarantee it'll work.
Longer term, we're hoping to support more systems "natively" including Mercurial, and possibly Git, so support may improve in the future.