On the one hand, it does seem like you still have far more chores than he does, even though he cooks and does the kitchen. But on the other hand, you can't even let his chores go. Why do you feel the need to micromanage him? That would drive me nuts and make me go toddler stroppy about not doing the jobs I'm being nagged about. Let him do it in his own time! And who in earth sweeps the kitchen floor in the evening?
I also think the issue with him not engaging with your DS is just different parenting. I spent my time as a SAHM mostly on MN. Engaging with toddlers just isn't something I enjoy, so I let mine get on with their own games, and while I did go to toddler groups, it was to get a break and a cup of tea with adults as much as for my DC.
The big problem here, it seems to me, is that you are the one required to do absolutely all the thinking about how the household is run, and that's absolutely exhausting. So you need to tell him bluntly, as I had to do with my DH at one point, that your mental health is suffering with the burdens you are carrying and you need him to step and up and take some of it off you. Point out all the "invisible" stuff you do in the background, the organising, the planning, the planning, and tell him you can't cope.
It has taken time and it does require you to step back, let go and trust your DH to be an adult like you and do the job - not necessarily in the way you would do it, but in his own way, which as also okay. You can't expect another adult to conform to your exacting views on the "right" way to do things. As frank said upthread, he's not a clone of you, he never will be and he is just as entitled to do things his way as you are to do them yours. He will never do things exactly the way you would, and to expect it will drive you crazy.
If he forgets things or moans that you didn't remind him or ask him, fix him with a Stare and say "I'm sorry, I mistook you for a competent responsible adult". That usually shames them into realising that they're being pathetic.
But you can't have it both ways. You can't expect your partner to do their fair share but maintain tight control.
Good luck!