Hi OP, be careful not to listen to too much of the negativity on here about how others would LTB immediately and he's got deep problems and he's not sorry. Those things are possible, but there may be other reasons. See below.
I've heard a lot of mums say they would have more time for each other when the children were older, but that's a unilateral decision made that affects two people. If the other person isn't content to be left without romance and connection for an extended period with no clear end in sight, then problems develop. I have a close friend who thought they would have time for each other when older, and they never had sex. Her husband didn't sleep with anyone else - confirmed by the OW when she messaged my friend on FB - but he went out looking for the emotional connection and romance that he couldn't get at home.
It's hard to advise, because only you know how disconnected the two of you have been.
If it was me, I wouldn't be able to get over cheating if we had a good relationship that included romance, emotional connection, sex, and date nights. If he cheated when things were fine, there's no fixing that.
But if I KNEW I had neglected his needs for emotional connection, romance, and sex, I wouldn't be so quick to LTB. People are only human and they do need some reason to live, someone to pay attention to them, closeness and fun that makes life seem worth living. And people vary hugely in how much they can take of a lack of connection. And also in how much connection they need.
Your husband seems to be doing and saying all the right things. However - and I know this will probably stick in the craw - you have to be willing to own whatever the extent of your neglect in the relationship is. I don't think this has to be the death knell of your relationship, but it's a warning that it's on oxygen. If you want it to work, you can't cast him in the role of villain long-term and you have to be willing to admit your own faults and commit to making changes. I would strongly suggest trying to get to the bottom of what made him step out, and for that, I think you need counselling. He says he doesn't know what he did it. He does know, but for whatever reason, he doesn't feel safe expressing his needs. Maybe he would in counselling.
He needs to commit to expressing whatever his needs are and in asking you to meet them, instead of looking at other people. And you need to be willing to hear him.
This is about communication.
I would absolutely do counselling, and I would not give up just yet on your marriage and family.