WIP: An (ongoing) experiment with adding native logs listener

[alwx]

📢 Type of change

• Bugfix
• New feature
• Enhancement
• Refactoring

📜 Description

💡 Motivation and Context

💚 How did you test it?

📝 Checklist

• I added tests to verify changes
• No new PII added or SDK only sends newly added PII if sendDefaultPII is enabled
• I updated the docs if needed.
• I updated the wizard if needed.
• All tests passing
• No breaking changes

🔮 Next steps

[github-actions]

Semver Impact of This PR

⚪ None (no version bump detected)

📋 Changelog Preview

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Entries from this PR are highlighted with a left border (blockquote style).

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• WIP: An (ongoing) experiment with adding native logs listener by alwx in #5622

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_{🤖 This preview updates automatically when you update the PR.}

[github-actions]

	Fails
🚫	Pull request is not ready for merge, please add the "ready-to-merge" label to the pull request
🚫	Please consider adding a changelog entry for the next release.

	Warnings
⚠️	🤖 Replay Stubs Check ✅ No changes detected in replay-stubs.jar All file contents are identical (verified by SHA-256 hash comparison).

Instructions and example for changelog

Please add an entry to CHANGELOG.md to the "Unreleased" section. Make sure the entry includes this PR's number.

Example:

## Unreleased

### Features

- An (ongoing) experiment with adding native logs listener ([#5622](https://github.com/getsentry/sentry-react-native/pull/5622))

If none of the above apply, you can opt out of this check by adding #skip-changelog to the PR description or adding a skip-changelog label.

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