I'm the gamer in the household but only in the evenings when the kids are in bed. The kids, being children, need their time monitoring, and while my eldest is mature and able to handle a couple of hours a day, my youngest tends to behave pretty poorly both in the game and off it, so she's off them for the foreseeable. It's all very controlled.
But yeah, I got tired of trying to get them out on walks so I booked myself on a couple of epic mountain hikes, adults only, and suddenly they seem a bit interested in it again now, and say they want to do 'cooler' walks too. So perhaps that worked. I will say that pursuing my own interests has been a successful tactic. Maybe kids like seeing their parents go off and have their own interests.
The issue here seems to be that your DH is keen to encourage this gamer lifestyle, which is unfortunate - he's an adult that's chosen it, having lived his life and done some other things, whereas your sons will not know anything else and will struggle to find meaning outside of an instant-gratification, immediate reward environment. They're being primed to only love one thing and will find it hard to expand their minds later. They will struggle to read, to be creative, things that take long hours. If they're playing shooters, they're also under a lot of adrenalin and hyper-focus - fun in short doses, but where little kids are concerned, it's again priming their system into one state for too long. It'll create hyper-vigilance and aggression later.
I'm not sure of any solutions, as often "talk to your husband" has been attempted before and, if it's done nothing, there's usually a risk of aggression in the response. Some DHs just won't be told.