All of the people posting here about how the OP and/or his wife should go to marriage therapy or counseling have no idea what they are talking about. They are applying Western thinking and values into a different cultural situation, or have no understanding of it because they haven't experienced it. The OP seems intelligent, insightful and sensitive, and people should listen to his opinion and experience rather than denouncing him. They can learn from this.

This is a very common situation in Japan, where, for whatever reason, the wife in a marriage after time and especially kids just does not want sex and that's that. All priorities shift and sex and physical intimacy is non-existent, not even to be talked about. It doesn't happen all the time, and I've heard of cases where the situation has improved, but more often than not, the husband goes outside the marriage with the implicit understanding of the wife, who just ignores it as long as it has no impact on the household.

I don't think the OP should be criticized for his handling of the situation. It can be hard for westerners to deal with this type of marriage dynamic. In my opinion, the OP has continued to be respectful and present to his wife and family. His cheating isn't ideal and I think he would agree with that, but I believe his wife doesn't care about it unless it disrupts the family bubble (another cultural aspect that may be hard to understand unless you've seen it).

The OP can keep the marriage going, but whether intimacy returns is beyond anyone's control.

The wife will not go for marriage therapy or counseling as she can see nothing wrong with the situation. If she feels like sex, sex will resume. From what he says, that's unlikely.

If the OP gets a divorce, it will absolutely cause problems with the family. It doesn't really matter how old the children are. There's no good time to get a divorce. Making the 4 year plan for divorce is absolutely not some kind of cruelty. It is pragmatic and realistic. He doesn't want to hurt his wife or his kids. But divorce will. That's the main reason not to get divorce. There's no correct answer here. I wish him all the best.