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.test.tsx

@@ -95,6 +95,24 @@ async function flushReact() {
 describe("Sidebar", () => {
   let container: HTMLDivElement;
 
+  async function renderSidebar() {
+    const root = createRoot(container);
+    const queryClient = new QueryClient({
+      defaultOptions: { queries: { retry: false } },
+    });
+
+    await act(async () => {
+      root.render(
+        <QueryClientProvider client={queryClient}>
+          <Sidebar />
+        </QueryClientProvider>,
+      );
+    });
+    await flushReact();
+
+    return root;
+  }
+
   beforeEach(() => {
     container = document.createElement("div");
     document.body.appendChild(container);
@@ -107,21 +125,23 @@ describe("Sidebar", () => {
     vi.clearAllMocks();
   });
 
-  it("does not flash the Workspaces link while experimental settings are loading", async () => {
-    mockInstanceSettingsApi.getExperimental.mockImplementation(() => new Promise(() => {}));
-    const root = createRoot(container);
-    const queryClient = new QueryClient({
-      defaultOptions: { queries: { retry: false } },
-    });
+  it("links the top search icon to the search page without showing Search in Work nav", async () => {
+    mockInstanceSettingsApi.getExperimental.mockResolvedValue({ enableIsolatedWorkspaces: false });
+    const root = await renderSidebar();
+
+    const topSearchLink = container.querySelector('a[aria-label="Search"]');
+    expect(topSearchLink?.getAttribute("href")).toBe("/search");
+    const workLinks = [...container.querySelectorAll("nav a")].map((anchor) => anchor.textContent?.trim());
+    expect(workLinks).not.toContain("Search");
 
     await act(async () => {
-      root.render(
-        <QueryClientProvider client={queryClient}>
-          <Sidebar />
-        </QueryClientProvider>,
-      );
+      root.unmount();
     });
-    await flushReact();
+  });
+
+  it("does not flash the Workspaces link while experimental settings are loading", async () => {
+    mockInstanceSettingsApi.getExperimental.mockImplementation(() => new Promise(() => {}));
+    const root = await renderSidebar();
 
     expect(container.textContent).not.toContain("Workspaces");
 
@@ -132,19 +152,7 @@ describe("Sidebar", () => {
 
   it("shows the Workspaces link when isolated workspaces are enabled", async () => {
     mockInstanceSettingsApi.getExperimental.mockResolvedValue({ enableIsolatedWorkspaces: true });
-    const root = createRoot(container);
-    const queryClient = new QueryClient({
-      defaultOptions: { queries: { retry: false } },
-    });
-
-    await act(async () => {
-      root.render(
-        <QueryClientProvider client={queryClient}>
-          <Sidebar />
-        </QueryClientProvider>,
-      );
-    });
-    await flushReact();
+    const root = await renderSidebar();
 
     const link = [...container.querySelectorAll("a")].find((anchor) => anchor.textContent === "Workspaces");
     expect(link?.getAttribute("href")).toBe("/workspaces");