refactor(logger): mark private members as readonly and fix code quality issues

[kawaaaas]

Summary

Changes

Please provide a summary of what's being changed

This PR improves code quality in the logger package by adding readonly modifiers to class members that are never reassigned after initialization, removing redundant code structures, and cleaning up unused collections in test files.

Note: The SonarQube issue AZItlgwecEUs2753LD68 for sampling.test.ts #sampleRateValue was not addressed as the relevant code was already removed in PR #4251.

Please add the issue number below, if no issue is present the PR might get blocked and not be reviewed

Issue number: closes #4255

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[boring-cyborg]

Thanks a lot for your first contribution! Please check out our contributing guidelines and don't hesitate to ask whatever you need.
In the meantime, check out the #typescript channel on our Powertools for AWS Lambda Discord: Invite link

[sonarqubecloud]

Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
0 Security Hotspots
0.0% Coverage on New Code
0.0% Duplication on New Code

See analysis details on SonarQube Cloud

[boring-cyborg]

Awesome work, congrats on your first merged pull request and thank you for helping improve everyone's experience!