The whole issue here seems to be Dad flapping around going 'what should I do?'
By his own admission he didn't do enough around the house. Mum took the lead with the kids and eventually she's reached a crunch point where something happened, Dad stopped backing her fully (note she was taking the lead on parenting still) and she couldn't take any more and has walked away.
Now Dad is still flapping and asking MN 'what should I do?' To which the answer is 'We don't know. You aren't communicating what has gone on. They are your kids and it's your wife. You need to step up and just get on with it instead of almost looking around for someone to approve what you are doing and take the lead for you with the kids'.
What strikes me, is the OP hasn't taken responsibility previously for the kids in this way. He's always followed Mum's lead and now he's somewhat floundering. It smells like she's almost been a parent to him as well as the kids, in taking on that role of leading the family.
It doesn't matter what's gone on before tbh. She's left. He can't change that. He can only control his own actions. He is going to have to start working out emotional responses by other by himself because that's that the role of parent. He needs to work on communicating better. He can't go asking MN to mindread because he can't do this communication.
I just get the sense of a sudden realisation that 'oh I might have to do this without being directed by my wife all the time. What do I do?' And somehow we will have magic solutions because we magically can mind read the situation.
Whilst MN is for support of parents there is a problem in that we can't offer that if someone doesn't want to engage enough to be able to do that. It leaves me with the impression that, this emotional checking out and dumping it all on the wife was probably the cause of the issue in the first place and unfortunately the situation now dictates this emotional hiding behind someone taking the lead is no longer an option. Basically the OP needs to step up and learn how to wing it as you go like every other parent.
It's tough, but there's nothing here from the information available that says anything different.