Selling the Xbox wasnt a smart move, kinda petty, and frankly pretty pointless, really.
Besides, if he’s not doing anything anyways, do you even need him there? If you’re doing everything by yourself with him, you may as well do everything by yourself without him. 
If he genuinely wants to be with you and the kids then really what needed to happen is some sort of serious conversation and/or couples therapy, and he needed to learn to grow up and realize that you aren’t a slave who exists to raise his children for him and take care of the house, alone, while he does nothing to help.
Even if he is the sole provider for the household, that does not excuse him from participating in and helping out with the household duties and the raising of your children. That’s absolute bs. I’ve worked 10-16 hour days in both physical labour jobs as well as extremely draining health care jobs, and I still managed to come home and make dinner and clean up around the house and parent my child; so I don’t buy the “I work for a living” excuse from anyone, in that regard. You live there, it’s your family, you’re responsible for taking care of things, too. Plain and simple.
Selling the Xbox wasn’t necessarily “right,” but I get how one can get to the end of their rope with a partner who really isn’t a partner at all; it’s honestly the most frustrating experience and can drive people mad. But yeah… despite that, it still wasn’t the right move. Not to mention, it was reactionary and impulsive, which never leads to good decisions.
But, maybe you should be asking yourself if it’s even worth it to you be with him at all, if his lack of participation in your relationship and the family you have together pushes you to the point of acting irrationally and makes you resort to things like selling his personal possessions while he’s at work. Neither of you are behaving in an emotionally mature manner; neither of you are contributing to improving your relationship with your actions. I mean, did selling his Xbox actually SOLVE anything? Or did it just create more problems on top of the ones you already weren’t dealing with?
Honestly, sounds like you’d both just be better off without each other. Sounds like an unhealthy relationship, all the way around.