🧇 Defer to component helpers to enable updated settings on Android

Deferring to `search_engines::ShouldShowUpdatedSettings()` matches the
behaviour of iOS and fixes a divergence in how the flags behave on
Android compared to Desktop as well.

Bug: b:315126434
Bug: b:318820137
Change-Id: I797993cf16c8884050fe881457e0a896b68d6cb6
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5169070
Commit-Queue: Nicolas Dossou-Gbété <dgn@chromium.org>
Code-Coverage: findit-for-me@appspot.gserviceaccount.com <findit-for-me@appspot.gserviceaccount.com>
Reviewed-by: Boris Sazonov <bsazonov@chromium.org>
Reviewed-by: Orin Jaworski <orinj@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1244616}

• chrome/android/javatests/src/org/chromium/chrome/browser/search_engines/TemplateUrlServiceTest.java[diff]
• chrome/android/javatests/src/org/chromium/chrome/browser/search_engines/settings/SearchEngineSettingsRenderTest.java[diff]
• chrome/android/junit/src/org/chromium/chrome/browser/search_engines/settings/SearchEngineAdapterTest.java[diff]
• chrome/browser/search_engines/android/java/src/org/chromium/chrome/browser/search_engines/settings/SearchEngineAdapter.java[diff]
• chrome/browser/search_engines/android/java/src/org/chromium/chrome/browser/search_engines/settings/SearchEngineSettings.java[diff]
• components/search_engines/android/java/src/org/chromium/components/search_engines/TemplateUrlService.java[diff]
• components/search_engines/android/template_url_service_android.cc[diff]
• components/search_engines/android/template_url_service_android.h[diff]
• components/search_engines/search_engine_choice_utils.cc[diff]
• components/search_engines/search_engine_choice_utils.h[diff]
• components/search_engines/search_engine_choice_utils_unittest.cc[diff]
• components/search_engines/template_url_service.cc[diff]
• components/search_engines/template_url_service.h[diff]