Maybe GitLab 15.4 (September 2022) will help:

Getting started with GitLab Pages just got easier

We’ve made it much easier to get started with GitLab Pages. Instead of creating configuration files by hand, build them interactively using the GitLab UI.
Just answer a few basic questions on how your app is built, and we’ll build the .gitlab-ci.yml file to get you started.

This is the first time we’re using our new Pipeline Wizard, a tool that makes it easy to create .gitlab-ci.yml files by building them in the GitLab UI.
You can look forward to more simplified onboarding helpers like this one.

See Documentation and Issue.

Plus, GitLab 17.8 (January 2025) adds:

Primary domain redirect for GitLab Pages

You can now set a primary domain in GitLab Pages to automatically redirect all requests from custom domains to your primary domain. This helps maintain SEO rankings and provides a consistent brand experience by directing visitors to your preferred domain, regardless of which URL they initially use to access your site.

See Documentation and Epic.

Ok, before ask, look at config files, my mistake.

There is no word about enabling Gitlab Pages in /etc/gitlab/gitlab.rb (in case of omnibus installs).

Current link to pages documentation: GitLab Pages

You do not have to create a new project if you already got one

The GitLab documentation is a little confusing about this. What the documentation means is you can get started with your Pages configuration by using an existing one of another template project. It does not mean you can only use GitLab Pages with new projects.

To use GitLab pages in your project, place a .gitlab-ci.yml in the root directory of your repository. Here, you can specify how the content of your GitLab Pages site should be generated and published, e.g.

pages:
  stage: deploy
  script:
  - mkdir .public
  - cp -r * .public
  - mv .public public
  artifacts:
    paths:
    - public
  only:
  - master

Source: https://about.gitlab.com/2016/04/07/gitlab-pages-setup/#add-gitlab-ci

TL,DR The GitLab Pages configuration is defined in your CI/CD configuration file (.gitlab-ci.yml). You can add or change this file at any point, not only when you create a new project.

Edit:
The published static content has to be in the public folder in the root of your project. There is no way around that. However, this is not a problem, you can simply move your stuff there during CI/CD, you do not need to change the repository content itself.

If you want to publish app/target/site/jacoco, you can do:

pages:
  stage: deploy
  script:
  - mv app/target/site/jacoco/ public
  artifacts:
    paths:
    - public
  only:
  - master