lingo I can identify with a lot of what you say here. It is a headfuck, as everyone seems to think they're an amazing, sensitive, New Age Man, but from the perspective of a partner, they're actually quite selfish and unempathetic.
My DP is similar, maybe a bit less frustrating in that he does have certain household chores that he knows are 'his share' and that he's expected to do, but he will barely ever see or do anything outside of that without me spelling it out to him/nagging. Often when I ask him to take care of something, he'll do part of the job, then leave the rest because he's "too busy" until I get fed up of waiting/asking and just do it myself.
He works a few more hours a week than me, but I do far and away the largest share of the housework and pretty much all of the mental load (which I don't think he even understands as 'a thing', even though I've tried to explain).
He writes revisionist histories of why I do what I do. For example, back in the pre-kids past, meal-planning was something we both did. Over time, whenever I asked him to contribute to the plan before we did the weekly shop, he was always too busy or didn't feel like thinking about it at that moment. He would do it later. Of course, he never did, and would get irritated if I asked again, or would say he would decide on the day (meaning he would end up going to the supermarket after work, come home late leaving me to deal with the whole bedtime-bathtime palaver, and we'd end up eating at 9pm. Or he would just get takeaways). So I ended up doing it all. Recently, when I complained about how this was yet another thing that had ended up on my plate because he tapped out, he said, in a surprised tone, "What?! You love meal planning, that's why you do it." 
If he has something on his mind, I'll be expected to give him my devoted attention and input for as long as required. If I've got something on my mind, he has a pretty limited tolerance for discussion. Conversely, sometimes he'll ask me for input on something, and if I give him suggestions he doesn't like, he'll suddenly be "too tired to talk" or "sick of talking about it."
Trying to talk about anything is such hard work. If I try to bring something up, it's always the wrong time to talk. If I say that there's something we need to talk about, he might agree, but I'll never be able to pin him down to a specific time, and he'll certainly never make an effort to bring it up again. I have to make all the effort. And given that part of the problem is that I'm already putting in the lion's share of the effort, knowing that I have to put in even more effort to try to get anything to change just seems too exhausting.