Yes you can turn AutoRestore On or Off using phpMyAdmin. The AutoRestore database option name and location are:
WordPress database table: xx_options
Column is option_name: bulletproof_security_options_ARCM
option value is: bps_autorestore_cron – the values are either On or Off (the first letter is capitalized)
What I suspect might have happened is that WordPress files were sent to Quarantine when you upgraded WordPress. That can happen depending on how the WordPress upgrade was done: ie something interfered with the AutoRestore Cron check, multiple things (WordPress, plugins, themes) were installed at the same time, the WordPress upgrade was installed remotely outside of WordPress. Check this AutoRestore/Quarantine log file: /wp-content/bps-backup/logs/autorestore_log.txt to see if that is what happened. Let me know if you see that WordPress files were quarantined.