Resolve "Set up filtering by site section"

What does this MR do and why?

Refactors filtering logic for Elasticsearch.

Closes #124 (closed)

Screenshots, screen recordings, or links to review app

These are strongly recommended to assist reviewers and reduce the time to merge your change.

Filter: Tutorial

Filter: User Docs

2 or more filters:

How to set up and validate locally

Numbered steps to set up and validate the change are strongly suggested.

1. Configure a local GitLab Docs environment: https://gitlab.com/gitlab-org/technical-writing-group/gitlab-docs-hugo/-/blob/main/doc/setup.md.
2. Search something on the search bar and hit enter to get into search results page.
3. Pick filters such as Tutorials, User Docs, etc. Make sure the results change and are relevant to filter AND make sure pagination is still working!

Merge request acceptance checklist

This checklist encourages us to confirm any changes have been analyzed to reduce risks in quality, performance, reliability, security, and maintainability.

• I have evaluated the MR acceptance checklist for this merge request.