You can't think it's for your girlfriend. Playing games is a waste of time. I don't have time to accompany you. You must respect your girlfriend, otherwise you two will definitely quarrel. I think a good boyfriend should understand his girlfriend's behavior. You can play a few games with her a little to coax her into enhancing her feelings. Don't regard playing games as an emotional obstacle. What really bothers you is definitely not your girlfriend playing games.
As a girl, I think playing games is nothing, and I can't understand why you want to stop her from playing games so much. Although the age difference is six years, I think you want the initiative more. Your girlfriend is older than you and has experienced more things than you. Being able to choose you six years younger than you proves that she recognizes you and thinks you can be kind to her. Even if you two are together, you should have your own preferences and personal space. It is wrong of you to interfere like this.
So don't try to stop your girlfriend from playing games. Playing games is also a way to relax as long as it doesn't delay normal work and life.