Convert sync_back.py to TypeScript

[mbg]

Follow-up (and based on) #3526.

This PR converts sync_back.py and test_sync_back.py to TypeScript equivalents.

Risk assessment

For internal use only. Please select the risk level of this change:

• Low risk: Changes are fully under feature flags, or have been fully tested and validated in pre-production environments and are highly observable, or are documentation or test only.

Which use cases does this change impact?

Environments:

• Testing/None - This change does not impact any CodeQL workflows in production.

How did/will you validate this change?

• Other - Please provide details.

If something goes wrong after this change is released, what are the mitigation and rollback strategies?

• Development/testing only - This change cannot cause any failures in production.

How will you know if something goes wrong after this change is released?

• Other - Please provide details.

Are there any special considerations for merging or releasing this change?

• No special considerations - This change can be merged at any time.

Merge / deployment checklist

• Confirm this change is backwards compatible with existing workflows.
• Consider adding a changelog entry for this change.
• Confirm the readme and docs have been updated if necessary.