An adult person with panic
disorder with agoraphobia is overwhelmed by anxiety specially triggered by leaving home, what could deeply undermine or limit his very ability for a normal job, social life, and other regular activities every adult needs to do outdoors in order to create a healthy and fulfilling life.
If there is no physical or mental disability affecting him, what he should be doing in order to develop a healthy life is mainly everything every adult needs to do within the limitations his anxiety disorder presents, taking full responsibility of his life, choices and actions. That would look like looking for a job he could perform from home. Nowadays there are many people who do work from home using phones, computers, emails, and many other forms of communication and technology. It does not matter what the activity but what is essential is for him to start living as a real adult and not as a disable person. In the mean time until he finds such a job, he should be focused on working with all necessary house chores a person his age could perform, thus he would work taking good care of personal needs and house chores in order to gain the right to receive housing and food while there, plus paying for his portions of bills.
Once he gets a job allowing him to have an income, then his activities at home should be adequately managed in order to take good care of such job, while not neglecting all his other basic personal responsibilities at home plus helping with house chores as time allows.
As with every healthy and responsible life style, he would have time to have fun, relax and do whatever he likes after complying with all his responsibilities, and that could include xbox or any other game, but again, in healthy responsible ways, as an adult and not as a spoiled child, even less as an addict, once the behavior you reported depicts a person addicted to such a game. If he happens to take full responsibility for his life, from bills to daily needs and then chooses to play 1, 2, 3 or more hours a day taking it from his sleep time, no problem, that'd be his choice, and he would be fully accountable for it, including all the consequences they create.
As you see, key factors here are responsibility, respect, daily productive life, collaboration with family responsibilities, among others. Family support, should become real "suppoort", meaning behaviors-decisions leading to promote his health, rehabilitation, accountability and maturity, and not any form of spoiling, codependent enabling of what is dysfunctional, unacceptable, addictive, neglectful or abusive.
I recommend family joining a support group for codependency and ideally to get family therapy in order to work on eradicating such addiction and to develop healthy ways to truly support your brother and cope with the challenges this situation presents.
In case you still have a tough time finding necessary local professional psychotherapeutic support for him, I would recommend considering online counseling as an alternative to get adequate initial support. Again, he is the only one with the power, so the responsibility to start and make his rehabilitation process work, but he needs healthy and assertive support. Any form of codependency would just continue to undermine every chance - effort for improvements here.
Hope it brings some light on this situation.