He may see the mess and then get overwhelmed by the enormity of it so he doesn't try
This is H. He hates messes, his mom is a hoarder and he has cleaned up after her in multiple homes - she feels that as long as there is a one-foot path not obscured by piles of stuff, the house if fine. He DOES have poor executive control and cannot manage time well many days. His cleaning have never been a day-to-day small cleaning to prevent a big huge effort being needed - his model, learned from his mom, was you let it go until something gets ruined or broken or a repairman is coming, and then you spend hours and hours making up for lost time.
But left alone to clean, he shuts down and can't do it. He finally admitted this in March - we had a day off work, and he wanted to get rid of clothes too big for him from losing weight. He has so many clothes that are too big, that fit, and are too small, because in his mind you hold on to them "just in case" your weight changes. He thinks he's saving money by not buying new clothes when his body changes, but we buy them anyway because he can't find anything that fits under all the clothes that don't.
So he agreed to clean out his closet, and I was going to go outside and take care of some stuff since the weather was good. He got quiet, and almost got frustrated, so I stopped, and asked, "do you want me to stay in here? I'm not sure how I can help, and I prefer to be alone when I am seeing if clothes fit - I figured you would like privacy." He admitted he cannot clean things away without me being in the room - it was too hard to do. So I stayed, and he filled up to laundry baskets of clothes he could bear to part with. Progress. It's taken us 21 years to get here, though.
H used to be at home all day. I worked. He'd not start cleaning until I got home, in which case, THEN he'd take out the trash, wipes some stuff, and even vacuum fur (we have cats) off the sofas. It was a big source of frustration for me for a long time and caused fights until I made peace with a few things myself.
1. My needs for cleanliness are different than his. I want it relatively clean all the time - he mostly worries about company coming or if things are in his way. He's fine washing a dish as he needs them. I want them washed and put away for when I need them. All couples have degrees of one being tidier than the other. I think you know how this stands in your relationship. I accepted I have a higher need for clean than H does, and with that comes a higher share of the work. If I want dishes done, and it makes me calm for themt o be done, I do them. If I need trash taken out, I do it.
2. If I lived alone - I'd do the same number of chores. There would be fewer dishes, towels, and the trash may fill up less often, but that's it. I'd do laundry, sweep and mop, dishes, grocery shopping, mow the lawn, weed eat, trash, pet care, home maintenance, with or without him. It's nice when he DOES take some initiative and do at least one chore. Honestly, when I am in my "mode" cleaning, he can get in my way trying to help, so I encourage him to do other things if I am cleaning for guests (prep for entertainment, stay in living room and "clean" in there by putting movies in order, etc.)
3. It's okay for things to be messy to a point. My bedroom gets the least amount of attention as a non-public space. It's clean - but my laundry is not folded. It's sitting on top of the basket, waiting to be put away. Dishes can sit by the sink for a day or two if I am tired, sick, or just plain busy. Things can pile up on the dining table for a few days, as long as I (and sometimes he) clear them off periodically to avoid having a huge pile of junk mail and odds and ends.
How old are the kids? Can they learn to be responsible for things like toys and their own trash? Can you make a "chore chart" for them (put your H ont here, too), to help them manage better? I know hainvg a messy home is stressful - it makes me feel anxious whent things hit a certain level of mess. But you would need baby steps to get your whole household to help out. It's not impossible, but it will take time to rewire everyone, including you, about wha'ts okay and what's not.