Some of this really resonates. For example, early on in our relationship, I asked DH to tidy the living room as it was a tip. He went in to do it and bizarrely decided to spend his time alphabetising the bookcase. 
That sparked a discussion about house chores and so we sat down (this is before kids) and compiled a list of all the house jobs that we agreed needed doing. Then we took it in turns to pick which ones we wanted to be responsible for, like picking for a sports team, until we had half each. This worked pretty well for a while.
Then kids happened and the inevitable DH working all the time and the mental load being passed to me. We still do struggle but he's more aware of it being an issue and we talk about it and try and find solutions.
We now each have delegated cooking nights so on his nights he needs to plan what to make, make sure the stuff is in the fridge for it, and cook it by a reasonable time so the toddler can eat it before bedtime. It does work now but when we started this system, any one of these steps could trip him up and it was bloody hard ignoring him and letting him get on with it instead of swooping in and just making DS some last minute emergency eggs on toast or something.
Part of it involves getting him into a routine so he doesn't have to think about it, rather than me project managing him. So he has reminders that he's set going off on his phone all the time to cook dinner/iron his shirts/take bins (recycling/non-recycling) out etc.
After a massive discussion about the roles we were falling into after I read the wifework book, he negotiated with work to cut down from FT slightly so he has 1 day each week at home with DS, when I work. He takes DS to playgroup where he chats to the other mums (of course they're all mums) and hears them complain about how crap their partners are at pulling their weight, and how frustrating it is, which I think helps him realise he doesn't want to be like that.
I'm currently 37 weeks pregnant, with bronchitis and a torn rib muscle, and despite all of the above we're still having to have discussions about him cleaning the bathroom, putting laundry on so we have clean towels and sheets, hoovering the house and taking over cooking on my nights.
I can't help thinking if it was him sick and pregnant I'd just be doing all that without needing to be managed. But then those jobs are in my head but not his. I need to find a way of getting them on his radar too. And it annoys me that even the task of getting them on his radar is my job. Grrr. Sorry for the long mildly ranty post.