If I understand you correctly, you're trying to get the Chrome app to update automatically, but this is failing.
Have you installed any anti-virus, clean-up or optimizing apps on your Mac? If so, uninstall them before proceeding. Also, disable any VPN software you might be using.
Then start by doing this:
• Install any updates that are available for your OS. If you're running Ventura, the current release is 11.6.3.
• Next, restart the Mac in Safe mode.Start up your Mac in safe mode - Apple Support
• While remaining in Safe mode, uninstall the Chrome app. Uninstall Google Chrome - Computer - support.google.com
• Empty the Trash, reboot the computer into Safe mode again and reinstall Chrome with a fresh download.
If the above doesn't appear to work, then this:
Please run an EtreCheck evaluation of your Mac and post the generated report back here in a reply. The report will not include any personal info. It simply details specifics about hardware performance and installed software that might be in conflict with the OS.
Please navigate to EtreCheck.com and download the free version. Be sure to Allow Full Disk Access when you install the app. Once you’ve run the app and created your report please post it with your reply to this message.
Please note that you must upload the full report. To see how, please click > How to use Add Text when posting… EtreCheck Report.
We’ll use your EtreCheck report to look for the things that may be causing your problem and advise how to correct them.