I assume you don't want to use GitLab shared runner which costs you and CI minutes limits.

Public folder

Refer to this gitlab-ci.yaml for making HTML content available https://gitlab.com/pages/plain-html/-/blob/master/.gitlab-ci.yml

Running locally without registration

Here is the known limitations: https://docs.gitlab.com/runner/commands/#limitations-of-gitlab-runner-exec

Pages are directly managed by Gitlab so running locally won't affect the page's state.

Use your own runner and register with gitlab (Possible)

Install GitLab runner on your machine and register as a runner, instructions can be found at Settings->CI/CD->Runners->Specific runner Enable it for your project or group.

I use my personal machine gitlab-runner to execute CI jobs that aren't counted towards CI minutes.

When creating a new GitLab project, GitLab allows you to start with a template instead of a blank project.
One of the templates is is the Pages/Plain HTML

This template uses a minimal CI workflow that does nothing.
All files in the public directory of the project will end up in the root of the web page.

For your use case, you just need to put your web page files (like index.html) into the public directory and run the CI pipeline once.
If your repository is on gitlab.com, one of the public runners can be used with your project without setting up any custom runners.

For more details, see the Create a GitLab Pages website from a project template documentation.

To view the logs of a GitLab Pipeline/Jobs go to Build -> Jobs
and click the latest Job with a red X symbol and Failed text.

The problem in your CI script is:

$ cp index.html public/
cp: cannot create regular file 'public/': Not a directory