The picker mounts whenever a cwd-aware builtin tool is enabled, so
it can open while file_glob_search is not served or was disabled by
the user. Every typed query then fired a search that could only
fail with a raw error.
Gate the debounced search on the tool state, the same way the
mention picker does, and show a message in place of the results
list that explains why search is unavailable. Manual entry with
Enter still commits a directory. The Browse button and the search
scope footer are hidden as well: Browse resolves the picked folder
name through file_glob_search, and the client-side toggle would not
stop that call.
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