My ex did something similar. I spoke to a solicitor and then told ex we needed to work out childcare arrangements in order to split assets and get divorced.
He then did get some advice himself and insisted on 50/50 split of childcare. Although he'd never done half of it before, his job did allow it and my solicitor said I couldn't fight it because he was in a position to do half. I thought he might change his mind when it became reality, but I think he was determined not to have to pay me any maintenance. We started by doing alternate weekends and I still did the weekday care. It took nine months for him to actually do 50/50. He said he wanted to, but he was quite reluctant in practice.
Then I also stopped doing any laundry etc for him. I cooked for him on my days in charge of the children. He sometimes cooked for me on his days in charge (it was often not appetising!)
All in all, it took two years to get him out of the house.
Even now, another 18 months on from him moving out, he will ask me to help him out by having the children more occasionally. For some reason, I manage to organise my life so that I don't need to ask him for the same in return. And I like it that way as I never wanted 50/50 so I am glad to see the DC more whenever possible.
I still occasionally find it completely baffling that although he left me, I was the one who made it happen. He never would have actually gone otherwise. Or maybe he would, but it would have taken forever.
For what it's worth, I was completely bereft and miserable and felt broken. But I am so much better off now (not financially, but mentally).
If he doesn't want a relationship any more then stop doing the things a partner would do.
My ex was a complete disaster with the DC at first. It was pretty awful at times (not dangerous or anything, but he wasn't very considerate towards them at times and it upset me to witness that). However, as far as I can tell, they are happy at his now. They want to see him and go happily. Children deserve a relationship with both their parents. Your H really needs to do some of that work and he will just have to learn how. My ex does seem to have changed his approach to parenting because he had to. It wasn't always working and he's had to adapt.