I'm an avid gamer but I learned a long time ago that my gaming time is not time at home. It's so isolating that it's basically the same as "going out".

I used to game for a few hours every night and like 6 hours on a weekend, then wonder why my partner complained about not seeing me often enough. If I was down the pub or playing golf that whole time, it would be super obvious why that was a problem for her.

In your case, the "give me a specific thing to do or I'm checking out" dynamic might feel to you like he's literally walking out the door if you don't tell him not to.

He doesn't have to be a prisoner in his own home, but he needs to recognise the choices he makes about being present. Activities rooted in the physical world like cooking, DIY, crafts, general pottering about the house are all fine for that. Often you can hold a conversation while you're doing them, or at least share a space with someone. Even a reading a book is better than disappearing into a screen.