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

tests/widget-system/page-display-save-harness.php

@@ -176,6 +176,7 @@ namespace {
 			'link' => ['url' => 'https://example.test/slide', 'target' => '_blank', 'text' => '<b>Go</b>', 'unknown' => 'drop-me'],
 			'text' => ['title' => '<strong>Safe</strong><script>bad()</script>', 'desc' => '<em>Copy</em>', 'unknown' => 'drop-me'],
 		]],
+		'header_slider_card_option' => ['show_widget_bg' => true, 'cards' => []],
 	];
 	$header_clean = $provider->sanitize_payload('page_display', $header_payload, 'header-components');
 
@@ -219,9 +220,10 @@ namespace {
 	 * @param list<string> $show_on
 	 * @param array<string,mixed> $page_config
 	 */
-	function aksrt_visibility_case(string $query, array $show_on, array $page_config = []): bool {
+	function aksrt_visibility_case(string $query, array $show_on, array $page_config = [], int $paged = 0): bool {
 		$GLOBALS['aksrt_visibility_query'] = $query;
 		$GLOBALS['aksrt_visibility_page_config'] = $page_config;
+		$GLOBALS['wp_query'] = new WP_Query(['paged' => $paged]);
 		\AKSRT\Settings\Settings::$headerSlider = [
 			'show_on_pages' => $show_on,
 			'show_type'     => '',
@@ -249,6 +251,7 @@ namespace {
 		'message_excludes_home'         => aksrt_visibility_case('message', ['home']),
 		'home_excludes_public_home'    => aksrt_visibility_case('user_home', ['home']),
 		'home_excludes_user_center'    => aksrt_visibility_case('user_center', ['home']),
+		'paged_archive_hidden'         => aksrt_visibility_case('home', ['home'], [], 2),
 	];
 
 	echo json_encode([
@@ -260,6 +263,7 @@ namespace {
 		'invalid_enum_rejected' => !$invalid_validation['valid'],
 		'nested_multiselect_valid' => $category_validation['valid'],
 		'nested' => $header_clean,
+		'header_card_bg' => $header_clean['header_slider_card_option']['show_widget_bg'] ?? null,
 		'footer_valid' => $footer_validation['valid'],
 		'footer_clean' => $footer_clean,
 		'footer_readback' => $footer_readback,