LightDrizzle is very astute:
I think if he comes back, you will have an angry, resentful bedfellow. He will feel humiliated that the children, friends and family know what he did, resentful that he has to settle for you, until the children are older, because his gold standard values prevent him just up and offing (his rationalisation). You haven’t even afforded him the dignity of pleading with him.
I think he will resent you, not feel grateful.
Should you both manage to muddle through, there is a huge risk of him leaving after the children reach independence and then you have lost the best part of a decade in which to reshape your life without him, and you will probably be financially worse off.
This is almost exactly what happened to me twenty odd years ago. He wasn't quite as bad as your H in that he admitted the affair - I was oblivious of course - and he had the protection of family and mutual friends not being aware of it. But yes, he wanted to leave only she changed her mind and he was distraught as he was in love with her. We tried separation but he would only fucking go for a couple of days before slinking back. He tried to pretend we had a discussion about what I wanted, (a separation) but he wouldn't comply, wouldn't move in with parents, said he could move into a cheap hotel - but he had no income and I couldn't afford the family mortgage plus a hotel for him.
As LightDrizzle predicts, he wore me down and I accepted him back and eight years later, when the kids were older teens, he fucked off anyway. I would have been much better off, emotionally, mentally and financially if he had gone when I wanted him to.
I wish MN had been around then - I wish I had confided in my parents (who would have supported me financially so I could tell him to fuck off), My life could have been SO different. I'm not unhappy now but I'm in my 60s and single and likely to remain so - and for the most part contentedly so I confess - but had he left when I was 40 I just feel I would have had a better chance at another relationship.