From the technical point of view: Move everything from the Game class that does not have to do with the QGraphicsView (point counting, time measurement, state management) into the new GameState class.
From the marketing point of view: Totally break an application that is working fine just to improve its interna.

From the personal point of view: It seems like I have an ambivalent relationship to refactoring object-oriented class layouts.

From the bugsquad point of view: PLEASE report anything that does not work.
I tried to ensure that anything continues to work after the transition, but I cannot be absolutely sure.