Notes isn't designed to be used for real-time collaborative work.
It isn't using the FaceTime connection to stream updates. What happens is that when you make a change to a Note, it is synched up to iCloud. At some point, your grandson's computer checks in with iCloud, sees the note has been updated and downloads those changes. And vice versa when he makes changes.
These updates are asynchronous - the Notes app periodically checks in with Apple's servers to push its own changes and see if there's an update from other users.
I can't speak for the increasing lag, other than I suspect that the app only checks after a certain idle period - the more you're using the app, the less frequently it checks in to see if others have made changes.
The adjunct to this is Freeform (included in the latest iOS and MacOS versions). Freeform is more of a collaborative whiteboard and is better suited to real-time collaboration (compared to Note's periodic synchronization). Try using Freeform to see if that works better.