Did you know GitLab also comes with a private container registry? https://about.gitlab.com/2016/05/23/gitlab-container-registr...
Without dind, it's really hard/annoying to us CI/CD to build docker images. But with Gitlab, they also provide a place to store docker images (the "registry") right next to your code, for free!
At work, we're investigating moving to Gitlab for everything except issues (which we'd need to keep on JIRA for now since we're so embedded with it). It looks like Gitlab does have some integration with JIRA, but it's project-level, and it would be nice if it could be group-level since we have many small repos. :)
(I don't have an affiliation with Gitlab, I'm just super happy with the service.)
We have a couple of open issues about integration with multiple JIRA projects: https://gitlab.com/gitlab-org/gitlab-ce/issues/25541 and https://gitlab.com/gitlab-org/gitlab-ce/issues/25758. Having group level integration can be a good solution for these requests. We will talk about it on those issues feel free to join the discussion.
That said, our (Codeship) Docker support is the most "Docker native" on the market, in my opinion. We build your containers, by default, using a Compose-based syntax and all commands are natively executed by your containers. There's no interacting with a Docker host or running explicit Docker commands at all.
We don't offer self-hosting but we do have a local CLI that lets you run and debug your process locally with parity to your remote builds.