[Backport] Extract system check rake task logic

What does this MR do?

Backport a refactor commit from https://gitlab.com/gitlab-org/gitlab-ee/merge_requests/8616.

These changes make the code more reusable, testable, and most importantly to the EE change, overrideable.

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
• Link to e2e tests MR added if this MR has Requires e2e tests label. See the Test Planning Process.