“Your game with foo has started!” notifications should be auto-deleted automatically after you’ve played your first move of that game (or otherwise have done some user action that demonstrates you already must be aware of the fact that the game has started).
“Your game with foo has ended!” notifications should only be sent out if it has been at last T time (e.g., 1 minute) since you played your last move of that game (otherwise you probably already know the result).
Reducing superfluous popups is an undisputible UI improvement, I’d say.
BTW --since this was brought up in the chat-- I’m willing to become a paid site supported, but only if OGS releases its source code under an open source license.