Gitlab and Element's website structure is different. element.io is the site for the managed product, so even their "on-premise" installer is meant to be used in a commercial relationship. For install docs you'd probably have to purchase their product (which isn't available freely).
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.