Fix context menu by filtering fake widgets starting with underscore (#1888)

The context menu was including fake widgets (like _omnibar-toggleAi for Duck.ai toggle)
which caused issues with native app integration. This change:

- Updates useContextMenu() to filter out all widgets with IDs starting with '_'
- Renames debug widget ID from 'debug' to '_debug' for consistency
- Adds integration test to verify only real widgets appear in context menu

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-authored-by: Claude <noreply@anthropic.com>

Author: 2 people

special-pages/pages/new-tab/app/customizer/components/CustomizerMenu.js

@@ -47,7 +47,7 @@ export function useContextMenu() {
             const items = getItems();
             /** @type {VisibilityMenuItem[]} */
             const simplified = items
-                .filter((x) => x.id !== 'debug')
+                .filter((x) => !x.id.startsWith('_'))
                 .map((item) => {
                     return {
                         id: item.id,

special-pages/pages/new-tab/app/omnibar/integration-tests/omnibar.spec.js

@@ -936,4 +936,25 @@ test.describe('omnibar widget', () => {
         await omnibar.closeButton().click();
         await omnibar.expectInputValue('');
     });
+
+    test('context menu only includes real widgets, not fake ones', async ({ page }, workerInfo) => {
+        const ntp = NewtabPage.create(page, workerInfo);
+        const omnibar = new OmnibarPage(ntp);
+
+        await ntp.reducedMotion();
+        await ntp.openPage({ additional: { omnibar: true } });
+        await omnibar.ready();
+
+        // Right-click on the page to trigger context menu
+        await page.click('body', { button: 'right' });
+
+        // Assert that contextMenu notification is sent with real widgets and not e.g. the Duck.ai toggle
+        await omnibar.expectMethodCalledWith('contextMenu', {
+            visibilityMenuItems: [
+                { id: 'omnibar', title: 'Search' },
+                { id: 'favorites', title: 'Favorites' },
+                { id: 'protections', title: 'Protections Report' },
+            ],
+        });
+    });
 });

special-pages/pages/new-tab/app/telemetry/Debug.js

@@ -10,7 +10,7 @@ export function DebugCustomized({ index, isOpenInitially = false }) {
     const telemetry = useTelemetry();
     useCustomizer({
         title: '🐞 Debug',
-        id: 'debug',
+        id: '_debug',
         icon: <DuckFoot />,
         visibility: isOpen ? 'visible' : 'hidden',
         toggle: (_id) => setOpen((prev) => !prev),