We have support for other flavors of Git on the roadmap, but for now we are focusing on providing the best possible experience with (public and private) GitHub repos.
Would love to see the API support a POST for creating a new build and passing in the project code (or slug url). This would enable the other git providers (Gitlab, BitBucket, Gogs, etc) and 3rd party services/frameworks (e.g. Factor.io) to provide the integration.
We do have an API that supports triggering builds with a POST request, passing in custom parameters, etc. The bigger issue is that we strive for a really rich integration between us and GitHub. For example, we support their status API, link back to PRs/commits from our build page, etc. It's a lot more than just triggering builds.
We do have support for other services planned, but we are focused on the GitHub experience for now.