What the?! I had no idea. My Mac running Ventura is NOT the Mac that stops syncing. In my situation it's the Mac still on Monterey but hopefully someone will post what the new calendarAgent is.
Looking at my Monterey Mac, CalendarAgent is an Executable in the Private Frameworks: /System/Library/PrivateFrameworks/CalendarAgent.framework
Looking through the various Frameworks in Ventura for the Calendar, I don't see any executables now so maybe that is why its process doesn't show up in Activity Monitor and of course isn't found via the Terminal command.
I doubt that it is the CalendarWidgetExtension or the com.apple.CalendarWeatherKitService. I'm not seeing any activity in Activity Monitor when adding and syncing events between the two computers.
BUT MAYBE this could be it?!
There is a CalendarDaemon now in the PrivateFrameworks and the process it runs is called "calaccessd"
I'm not sure launchctl to stop and run should be calaccessd (/System/Library/PrivateFrameworks/CalendarDaemon.framework/Support/calaccessd)
But I think that is likely the best place to start.