Rework the search field behavior in the sidebar

We just tested it with another user an noticed some more issues:
* If the user is on the search route it should be reflected in the sidebar
* Clicking on the search icon alone should lead to the search page

plugins/search/src/components/SidebarSearch/SidebarSearch.tsx

@@ -31,5 +31,5 @@ export const SidebarSearch = () => {
     [navigate],
   );
 
-  return <SidebarSearchField onSearch={handleSearch} />;
+  return <SidebarSearchField onSearch={handleSearch} to="/search" />;
 };