Pass filter value when clicking Browse All in Nav Editor screen

[getdave]

Let's port over change from #34912 which allows the search value to be passed over to the inserter when "Browse All" is triggered. THis saves the user from having to re-type their search query.

[getdave]

Add to the Tracking Issue under Normal.

[kathrynwp]

Closing this issue due to the Navigation Screen project being moved to an inactive status on the feature projects page in coordination with the project leads. (The developer documentation in the initial post are no longer accessible)

If this work is picked back up, issues can be revisited and reopened as needed. Thanks for pitching in on this early feature so the wider WordPress project could benefit from the lessons learned!