Add basic_search parameter to search API

What does this MR do?

This MR adds basic_search parameter to Search API. It's already accessible via web search controller.

This MR doesn't have a changelog entry since this is not a user facing change and we only use this parameter for debugging and performance benchmarking purposes.

Screenshots (strongly suggested)

Does this MR meet the acceptance criteria?

Conformity

• [-] Changelog entry
• Documentation (if required)
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content

Availability and Testing

• Review and add/update tests for this feature/bug. Consider all test levels. See the Test Planning Process.
• Tested in all supported browsers
• Informed Infrastructure department of a default or new setting change, if applicable per definition of done

Security

If this MR contains changes to processing or storing of credentials or tokens, authorization and authentication methods and other items described in the security review guidelines:

• Label as security and @ mention @gitlab-com/gl-security/appsec
• The MR includes necessary changes to maintain consistency between UI, API, email, or other methods
• Security reports checked/validated by a reviewer from the AppSec team