Element: landing page https://element.io/solutions/self-hosted-or-cloud-collaborat...
GitLab: all the things https://about.gitlab.com/install/ at the bottom is the Debian package, but if you go to the Docker Page and click the username you can see gitlab-ce there as well.
You probably want to read the open-source software's installation instructions at: https://github.com/vector-im/element-web
TLDR, check out the project, run `yarn install`, then edit the config file, then `yarn build`.
And, yes, that is all there is to it. It's significantly simpler to deploy than GitLab.
Finally, you keep mentioning self-hosting; you _can_ just use a non-self hosted application like the downloadable version of Element, SchildiChat, Fluffychat, or any other client.
No reason to bring hosting into the mix for the client, if that's causing concern.
To me it looks like they come with proprietary stuff: https://element.io/pricing
Especially Group Sync seems like something to drive you into their paid offerings, I assume by keeping the code close to the vest.
You can host bridges yourself too
You yourself say in a different comment that you are currently assessing Element: https://news.ycombinator.com/item?id=34779070. Either you have a formed opinion Matrix, or you don't. Which one is it?
You keep saying Element instead of Matrix, and obviating the whole Matrix ecosystem. Matrix protocol has several server implementations, and many more things around: https://matrix.org/docs/projects/try-matrix-now
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> The Dockerfile used for building public images is in Omnibus Repository