[Android] Fix Clear data on exit not clearing omnibox suggestions (#35790)

ShortcutsBackend initializes asynchronously. When clearBrowsingData() was
called during startup, the backend was still in the INITIALIZING state and
its OnHistoryDeletions handler silently returned early, leaving shortcut
suggestions (clock-icon entries) intact after exit.

Fix by adding a JNI bridge (BraveShortcutsUtils.initThenRun) that waits for
ShortcutsBackend to finish its async DB init via OnShortcutsLoaded() before
invoking clearBrowsingData(). Also move the clear-on-exit call from
initializeState() to finishNativeInitialization() where the profile is
available, and add FORM_DATA (autofill) to the cleared data types.

browser/android/browsing_data/BUILD.gn

@@ -0,0 +1,19 @@
+# Copyright (c) 2026 The Brave Authors. All rights reserved.
+# This Source Code Form is subject to the terms of the Mozilla Public
+# License, v. 2.0. If a copy of the MPL was not distributed with this file,
+# You can obtain one at https://mozilla.org/MPL/2.0/.
+
+assert(is_android)
+
+source_set("browsing_data") {
+  sources = [ "brave_shortcuts_utils.cc" ]
+
+  deps = [
+    "//base",
+    "//chrome/android:chrome_jni_headers",
+    "//chrome/browser/autocomplete",
+    "//chrome/browser/profiles:profile",
+    "//components/omnibox/browser",
+    "//third_party/jni_zero",
+  ]
+}

browser/android/browsing_data/brave_shortcuts_utils.cc

@@ -0,0 +1,64 @@
+/* Copyright (c) 2026 The Brave Authors. All rights reserved.
+ * This Source Code Form is subject to the terms of the Mozilla Public
+ * License, v. 2.0. If a copy of the MPL was not distributed with this file,
+ * You can obtain one at https://mozilla.org/MPL/2.0/. */
+
+#include <jni.h>
+
+#include "base/android/scoped_java_ref.h"
+#include "chrome/android/chrome_jni_headers/BraveShortcutsUtils_jni.h"
+#include "chrome/browser/autocomplete/shortcuts_backend_factory.h"
+#include "chrome/browser/profiles/profile.h"
+#include "components/omnibox/browser/shortcuts_backend.h"
+#include "third_party/jni_zero/common_apis.h"
+
+namespace brave {
+
+namespace {
+
+// One-shot observer: waits for ShortcutsBackend to finish its async DB init,
+// then runs the Java callback (which calls clearBrowsingData). At that point
+// OnHistoryDeletions will see an initialized backend and handle shortcuts
+// correctly without any extra workarounds.
+class ShortcutsInitCallbackObserver
+    : public ShortcutsBackend::ShortcutsBackendObserver {
+ public:
+  ShortcutsInitCallbackObserver(
+      scoped_refptr<ShortcutsBackend> backend,
+      base::android::ScopedJavaGlobalRef<jobject> j_callback)
+      : backend_(std::move(backend)), j_callback_(std::move(j_callback)) {
+    backend_->AddObserver(this);
+  }
+  ~ShortcutsInitCallbackObserver() override { backend_->RemoveObserver(this); }
+
+  void OnShortcutsLoaded() override {
+    jni_zero::RunRunnable(j_callback_);
+    delete this;
+  }
+  void OnShortcutsChanged() override {}
+
+ private:
+  scoped_refptr<ShortcutsBackend> backend_;
+  base::android::ScopedJavaGlobalRef<jobject> j_callback_;
+};
+
+}  // namespace
+
+void JNI_BraveShortcutsUtils_InitThenRun(
+    JNIEnv* env,
+    const jni_zero::JavaRef<jobject>& j_profile,
+    const jni_zero::JavaRef<jobject>& j_callback) {
+  Profile* profile = Profile::FromJavaObject(j_profile);
+  auto backend = ShortcutsBackendFactory::GetForProfile(profile);
+  if (!backend || backend->initialized()) {
+    jni_zero::RunRunnable(j_callback);
+    return;
+  }
+  new ShortcutsInitCallbackObserver(
+      std::move(backend),
+      base::android::ScopedJavaGlobalRef<jobject>(j_callback));
+}
+
+}  // namespace brave
+
+DEFINE_JNI(BraveShortcutsUtils)