Resolve "`backup_json` log is missing"

What does this MR do and why?

Restores creation of backup_json.log file. This was inadvertently disabled as part of a code refactor

MR acceptance checklist

Please evaluate this MR against the MR acceptance checklist. It helps you analyze changes to reduce risks in quality, performance, reliability, security, and maintainability.

How to set up and validate locally

Numbered steps to set up and validate the change are strongly suggested.

1. In GDK run bin/rake gitlab:backup:create
2. Currently, output will be logged to stdout only
3. Checkout this branch, and re-run bin/rake gitlab:backup:create
4. Output should be logged to stdout, and logs/backup_json.log

Related to #464028 (closed)