Add debug logging to synology_dsm

[mib1185]

Proposed change

Add debug logging output for better troubleshooting in case of issues.
Need rebase as soon as #42697 and #42777 are merged.
Opened as draft to collect, if existing, additional suggestions.

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:
• 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.

If user exposed functionality or configuration variables are added/changed:

• Documentation added/updated for www.home-assistant.io

If the code communicates with devices, web services, or third-party tools:

• The manifest file has all fields filled out correctly.
  Updated and included derived files by running: python3 -m script.hassfest.
• New or updated dependencies have been added to requirements_all.txt.
  Updated by running python3 -m script.gen_requirements_all.
• Untested files have been added to .coveragerc.

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• I have reviewed two other open pull requests in this repository.

[probot-home-assistant]

Hey there @hacf-fr, @Quentame, mind taking a look at this pull request as its been labeled with an integration (synology_dsm) you are listed as a codeowner for? Thanks!
_{^{(message by CodeOwnersMention)}}

[github-actions]

There hasn't been any activity on this pull request recently. This pull request has been automatically marked as stale because of that and will be closed if no further activity occurs within 7 days.
Thank you for your contributions.

[frenck]

This PR has been in a draft and running stale without activity since November.
I'll go ahead and close this PR for now. Feel free to re-open the PR again when ready to start working on it. Contributions are always appreciated 👍