[cr137] ExtensionBrowserTest::observer_ moved to base class

This change adjusts how we are accessing these observers, as the field
has been moved to a base class.

Chromium changes:
https://chromium.googlesource.com/chromium/src/+/f41b0b2533c8f2c870d26dffe97eb96809eba668

commit f41b0b2533c8f2c870d26dffe97eb96809eba668
Author: Devlin Cronin <rdevlin.cronin@chromium.org>
Date:   Mon Apr 7 16:19:25 2025 -0700

    [Extensions] Hoist observer_ to ExtensionPlatformBrowserTest

    ExtensionBrowserTest has a ChromeExtensionTestNotificationObserver
    member, `observer_`, which is used to wait for various extension-
    related events. ChromeExtensionTestNotificationObserver can't be
    included in the experimental desktop android build yet, but the
    parent class, ExtensionTestNotificationObserver, can.

    Add a `test_notification_observer_` member on
    ExtensionPlatformBrowserTest that's implemented as an
    ExtensionTestNotificationObserver by default, and overridden to the
    ChromeExtensionTestNotificationObserver in ExtensionBrowserTest. This
    allows us to move `ExtensionBrowserTest::WaitForExtensionViewsToLoad()`
    up to ExtensionPlatformBrowserTest and will enable more methods to move
    up in the future.

    This CL has no effective behavior change.

    Bug: 404581990
    Change-Id: I252419e1ac891e993ba74df3ca922376ab7704aa
    Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6435314
    Reviewed-by: James Cook <jamescook@chromium.org>
    Commit-Queue: Devlin Cronin <rdevlin.cronin@chromium.org>
    Cr-Commit-Position: refs/heads/main@{#1443826}

browser/extensions/brave_extension_functional_test.cc

@@ -35,7 +35,7 @@ ExtensionFunctionalTest::InstallExtensionSilently(
 
   installer->InstallCrx(path);
   EXPECT_TRUE(registry_observer.WaitForExtensionInstalled());
-  EXPECT_TRUE(observer_->WaitForExtensionViewsToLoad());
+  EXPECT_TRUE(test_notification_observer()->WaitForExtensionViewsToLoad());
 
   size_t num_after = registry->enabled_extensions().size();
   EXPECT_EQ(num_before + 1, num_after);