fix(frontend): align search and header components with reference

tests/widget-system/page-display-contract.test.cjs

@@ -447,6 +447,7 @@ test('save contract rejects invalid input and round-trips canonical nested value
   assert.equal(data.unknown_section, true)
   assert.equal(data.invalid_enum_rejected, true)
   assert.equal(data.nested_multiselect_valid, true)
+  assert.equal(data.header_card_bg, true)
   assert.equal(data.footer_valid, true)
   assert.equal(data.vip_switches_valid, true)
   assert.deepEqual(data.vip_switches_clean, {
@@ -495,5 +496,6 @@ test('save contract rejects invalid input and round-trips canonical nested value
     message_excludes_home: false,
     home_excludes_public_home: false,
     home_excludes_user_center: false,
+    paged_archive_hidden: false,
   })
 })