Even with more zen type mediation where you try to empty your mind, it's still very different from a game flow state. The entire point of concentrating on your breathing is that your breathing is always there and is boring!
Once you reach some level of mindfulness, you start to notice that your breath is very seldom there. There are all kinds of distractions that your attention is moving to. You may not notice it and 'think' that you are watching the breath, while in the center of your attention is eg. an internal monologue or concept of the breath or mental image of the breath (how you imagine it).
> The entire point of concentrating on your breathing is that your breathing is always there and is boring!
Here you go. If there's boredom, there's no breath. If your attention is on the breath there's no boredom. Boredom is a specific mental object (state and story) and it's a distraction from the breath. 'Breath' is a concept and looking at a concept is boring indeed. If you look at the actual breath, however, you see something that's never the same, constantly changing stream of sensations, reality behind the concepts. It's always amazing, it's more interesting than anything you can ever see in ordinary closed, story-driven state of mind.
Concentrating on the breath is an excellent entry point to a set of much deeper and more interesting experiences.
A video game is not going to lead you to these, because it is too distracting/complex/engaging. The breath can. Boring was perhaps not a good way to express this, but it was simple, without the pull of these subtler sensations being noticeable there is no path forward.
This stream is exponentially less noticeable while doing something less boring, like gaming, to a beginner.
Perhaps "stimulating" is a better concept than "boring" to explain it?
It is something available to all beginners at all times and works well as a place to refocus. And at that stage the main challenge is stopping constant mind wandering and narrativising.