Plex tests cleanup and additions

[jjlawren]

Proposed change

• Remove commented out tests which were failing in 3.8 (Use built-in test helpers on 3.8 #34901).
• Add tests to complete test coverage for unskipped files.
• Remove unnecessary method found during testing.

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New integration (thank you!)
• New feature (which adds functionality to an existing integration)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

Additional information

• This PR fixes or closes issue: fixes #
• This PR is related to issue: Use built-in test helpers on 3.8 #34901
• Link to documentation pull request:

Checklist

• The code change is tested and works locally.
• Local tests pass. Your PR cannot be merged unless tests pass
• There is no commented out code in this PR.
• I have followed the development checklist
• The code has been formatted using Black (black --fast homeassistant tests)
• Tests have been added to verify that the new code works.

The integration reached or maintains the following Integration Quality Scale:

• No score or internal
• 🥈 Silver
• 🥇 Gold
• 🏆 Platinum

[MartinHjelmare]

It would be better to patch the websocket and get access to the callback via the patch call and then call the callback to trigger an update. We don't want to use integration details like this in the tests.

[jjlawren]

Done in #37147