Refs #35514 -- Cleaned up logging and middleware tests.

Simplified and removed use of soon-to-be-deprecated or removed features
from tests for AdminEmailHandler and BrokenLinkEmailsMiddleware. Added
or updated cases to ensure `fail_silently` usage remains consistent
during the transition to email providers.

Changed custom FailingEmailBackend to respect fail_silently and record
its use in a class-level property.

In AdminEmailHandlerTest:

* Added make_log_record() helper to reduce duplicated code.

* Changed test_fail_silently() to send through the FailingEmailBackend,
  which will fail if not initialized with fail_silently=True. (Rather
  than inspecting attributes of a soon-to-be-deprecated connection.)

* Simplified test_uses_custom_email_backend() by replacing monkeypatches
  with inspectable custom backend.

* In test_customize_send_mail_method(), removed use of the undocumented
  self.connection() method and the soon-to-be-deprecated `connection`
  arg to mail_managers(). This test verifies a subclass can customize
  sending by mailing the managers rather than the admins. Involving a
  email backend connection was irrelevant.

In BrokenLinkEmailsMiddlewareTest, added a case with FailingEmailBackend
to verify send uses fail_silently=True.

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