Horrible situation. All I can say is that he should not move out. If she wants to leave, that’s her choice. He should stay and try to keep the kids.
I agree that the dad often gets screwed in these situations. I haven’t been thru this, so my advice is somewhat empty. But if he can be a “chump” for his kids sake, it may be better.
Sounds like she would like to keep this “marriage in name only” for convenience’ sake.
It would be better for the boys to stay with him and for her to go. Girls would be the opposite situation. He could offer to pay her a lump sum to just go, then he would have no more need to fume. Let the new guy have the trash on a silver platter.