[compat] Ensure submodules are imported correctly

Closes #7663

yt_dlp/compat/__init__.py

@@ -1,14 +1,11 @@
 import os
 import sys
-import warnings
 import xml.etree.ElementTree as etree
 
-from ._deprecated import *  # noqa: F401, F403
 from .compat_utils import passthrough_module
 
-# XXX: Implement this the same way as other DeprecationWarnings without circular import
-passthrough_module(__name__, '._legacy', callback=lambda attr: warnings.warn(
-    DeprecationWarning(f'{__name__}.{attr} is deprecated'), stacklevel=5))
+passthrough_module(__name__, '._deprecated')
+del passthrough_module
 
 
 # HTMLParseError has been deprecated in Python 3.3 and removed in

yt_dlp/compat/_deprecated.py

@@ -1,4 +1,12 @@
 """Deprecated - New code should avoid these"""
+import warnings
+
+from .compat_utils import passthrough_module
+
+# XXX: Implement this the same way as other DeprecationWarnings without circular import
+passthrough_module(__name__, '.._legacy', callback=lambda attr: warnings.warn(
+    DeprecationWarning(f'{__name__}.{attr} is deprecated'), stacklevel=6))
+del passthrough_module
 
 import base64
 import urllib.error

yt_dlp/compat/urllib/__init__.py

@@ -1,6 +1,9 @@
 # flake8: noqa: F405
 from urllib import *  # noqa: F403
 
+del request
+from . import request  # noqa: F401
+
 from ..compat_utils import passthrough_module
 
 passthrough_module(__name__, 'urllib')