keeping the search field in the toolbar has several benefits:
While some of the benefits can be reproduced even if the search field is placed next to the navigation panel, the user customizability is definitely lost which is not something I'm keen on giving up.
Given that the Search field affects the contents of the Navigation Panel, I'd like to see it moved from the Toolbar. Ideally if would be pinned to the top of the Navigation panel.
Doing so would provide a visual and mental link between the two areas.