[QA] Improve internals of hashed storage settings-related files

What does this MR do?

This improves the architecture/internals after https://gitlab.com/gitlab-org/gitlab-ee/merge_requests/7531.

This takes some changes from https://gitlab.com/gitlab-org/gitlab-ee/merge_requests/7601.

What are the relevant issue numbers?

Does this MR meet the acceptance criteria?

• [-] Changelog entry added, if necessary
• [-] Documentation created/updated
• [-] Tests added for this feature/bug
• Conforms to the code review guidelines
• Conforms to the merge request performance guidelines
• Conforms to the style guides
• Conforms to the database guides
• EE specific content should be in the top level /ee folder
• [-] For a paid feature, have we considered GitLab.com plans, how it works for groups, and is there a design for promoting it to users who aren't on the correct plan?

Closes #7763 (closed)