Route sidebar search icon directly to search (#5440)

## Thinking Path

> - Paperclip orchestrates AI agents for zero-human companies
> - Operators use the sidebar as their primary board navigation surface
> - The board now has a dedicated search page, so the header search icon
should behave as normal navigation instead of only dispatching a
command-palette shortcut
> - The Work nav also had a separate Search row, which duplicated the
always-visible header search affordance
> - This pull request keeps search one click away while making it a
direct `/search` link and reducing sidebar nav noise
> - The benefit is a smaller, clearer sidebar with search still
accessible from the top-level chrome

## What Changed

- Changed the sidebar header search icon into a direct `NavLink` to
`/search`.
- Removed the duplicate `Search` row from the Work navigation section.
- Added focused Sidebar coverage that asserts the header search link
target and confirms Search is not rendered in the Work nav.
- Refactored the Sidebar test setup helper to avoid repeating the React
Query wrapper across tests.

## Verification

- `pnpm install --frozen-lockfile` in the PR worktree so workspace
package symlinks existed for test execution. This completed with
existing plugin SDK bin warnings for missing built artifacts.
- `pnpm exec vitest run ui/src/components/Sidebar.test.tsx` — 3 passed.
- `pnpm --filter @paperclipai/ui typecheck` — passed.

## Risks

- Low: this changes a sidebar navigation affordance only. Users who
previously clicked the header icon now land on the full search page
instead of opening the command-palette shortcut path.
- Low: removing the Work nav Search row could affect users who expected
Search in that section, but the icon remains in the fixed sidebar header
and is covered by a targeted DOM test.

> For core feature work, check [`ROADMAP.md`](ROADMAP.md) first and
discuss it in `#dev` before opening the PR. Feature PRs that overlap
with planned core work may need to be redirected — check the roadmap
first. See `CONTRIBUTING.md`.

## Model Used

- OpenAI Codex coding agent, GPT-5 model family (`gpt-5`), tool-enabled
Paperclip heartbeat environment. Context window and internal reasoning
mode are not exposed by the runtime.

## Checklist

- [x] I have included a thinking path that traces from project context
to this change
- [x] I have specified the model used (with version and capability
details)
- [x] I have checked ROADMAP.md and confirmed this PR does not duplicate
planned core work
- [x] I have run tests locally and they pass
- [x] I have added or updated tests where applicable
- [x] If this change affects the UI, I have included before/after
screenshots or equivalent focused UI verification
- [x] I have updated relevant documentation to reflect my changes
- [x] I have considered and documented any risks above
- [x] I will address all Greptile and reviewer comments before
requesting merge

---------

Co-authored-by: Paperclip <noreply@paperclip.ing>

ui/src/components/Sidebar.tsx

@@ -14,6 +14,7 @@ import {
   Settings,
 } from "lucide-react";
 import { useQuery } from "@tanstack/react-query";
+import { NavLink } from "@/lib/router";
 import { SidebarSection } from "./SidebarSection";
 import { SidebarNavItem } from "./SidebarNavItem";
 import { SidebarProjects } from "./SidebarProjects";
@@ -45,10 +46,6 @@ export function Sidebar() {
   const liveRunCount = liveRuns?.length ?? 0;
   const showWorkspacesLink = experimentalSettings?.enableIsolatedWorkspaces === true;
 
-  function openSearch() {
-    document.dispatchEvent(new KeyboardEvent("keydown", { key: "k", metaKey: true }));
-  }
-
   const pluginContext = {
     companyId: selectedCompanyId,
     companyPrefix: selectedCompany?.issuePrefix ?? null,
@@ -60,12 +57,16 @@ export function Sidebar() {
       <div className="flex items-center gap-1 px-3 h-12 shrink-0">
         <SidebarCompanyMenu />
         <Button
+          asChild
           variant="ghost"
           size="icon-sm"
           className="text-muted-foreground shrink-0"
-          onClick={openSearch}
+          aria-label="Search"
+          title="Search"
         >
-          <Search className="h-4 w-4" />
+          <NavLink to="/search">
+            <Search className="h-4 w-4" />
+          </NavLink>
         </Button>
       </div>
 
@@ -99,7 +100,6 @@ export function Sidebar() {
 
         <SidebarSection label="Work">
           <SidebarNavItem to="/issues" label="Issues" icon={CircleDot} />
-          <SidebarNavItem to="/search" label="Search" icon={Search} />
           <SidebarNavItem to="/routines" label="Routines" icon={Repeat} />
           <SidebarNavItem to="/goals" label="Goals" icon={Target} />
           {showWorkspacesLink ? (