[ie/youtube] Add web_embedded client for age-restricted videos (#13089)

Authored by: bashonly

yt_dlp/extractor/youtube/_video.py

@@ -3115,9 +3115,19 @@ class YoutubeIE(YoutubeBaseInfoExtractor):
                 else:
                     prs.append(pr)
 
+            # web_embedded can work around age-gate and age-verification for some embeddable videos
+            if self._is_agegated(pr) and variant != 'web_embedded':
+                append_client(f'web_embedded.{base_client}')
+            # Unauthenticated users will only get web_embedded client formats if age-gated
+            if self._is_agegated(pr) and not self.is_authenticated:
+                self.to_screen(
+                    f'{video_id}: This video is age-restricted; some formats may be missing '
+                    f'without authentication. {self._youtube_login_hint}', only_once=True)
+
             # EU countries require age-verification for accounts to access age-restricted videos
             # If account is not age-verified, _is_agegated() will be truthy for non-embedded clients
-            if self.is_authenticated and self._is_agegated(pr):
+            embedding_is_disabled = variant == 'web_embedded' and self._is_unplayable(pr)
+            if self.is_authenticated and (self._is_agegated(pr) or embedding_is_disabled):
                 self.to_screen(
                     f'{video_id}: This video is age-restricted and YouTube is requiring '
                     'account age-verification; some formats may be missing', only_once=True)