The documented URL scheme for search was designed for use with Shortcuts - it provides the results as JSON which can then be parsed and displayed.
There is however an undocumented URL scheme which is used by the widget which may work for you. It is almost exactly as you've provided however the path is tasks instead of search. For example beorg://x-callback-url/tasks?search=drive.
I've added a task for later this week to add this to the documentation in case anyone else is looking for this capability. Thanks for your question!