FWIW, Pierre's "Code Storage" project [1] seems like it simplifies a lot of the operational overhead of running git servers, if what you want is "an API for git push". Not affiliated with the company (and I haven't tried it myself, so I can't vouch for how well it works), I just think it's a neat idea.
I think "Code Storage" (definitely needs a unique name), is less an API for git push (surely git push is that API?), and more an API for "git init"? It seems to be Git as infrastructure, rather than Git as a product. i.e. if you're using it for a single repo it's probably not a good fit, it's for products that themselves provide git repos.