Thank you for reaching out to us. Search page uses a special template just like the Blog or Archives pages and cannot be edited using a content builder / Cornerstone. That is how it works by default on WordPress. Depending on the layout you want to achieve, for simple changes, it might be possible with custom CSS (you can follow Change search page layout)
For complex changes, customizing a template would be needed and will be outside the scope of our support.
Please note that custom coding is outside the scope of our support. Issues that might arise from the use of custom code in the above thread and further enhancements should be directed to a third party developer.
Hope this helps!