I would expect him to do the admin and anything to do with school. After all, he is there and knows what's going on where you do not. Banking stuff etc is easier to do by the person there in the day. Cleaning, otoh, can be done whenever by whomever and I don't think it's an exclusive housewife/husband job.
Do you think it's more of a question of organisation/motivation? If I don't get started on something straightaway, everything goes to pot. It literally throws the whole day out. Both of mine are at school and, on my days off, say, if I have an appointment (hair/dentist/doctor/hospital) smack bang in the middle of the day (I try to book appointments early in the morning), there's not much I can get done either side. Likewise, if I have to go into town, it easily uses up 1.5 to 2 hours when you include driving there, finding a space, traffic, queues and finding what I need and other things popping up which I might as well get while I am there.
When DD was at pre-school, I worked on the principle of one job per day. So, one day would be ironing, one might be cleaning sinks/loos. The rest, I'd keep ticking over - laundry, cooking, cleaning the kitchen would get done all the time. When DD was home, I would take her shopping as she was OK and liked it (could not do this with DS). I certainly could not do anything big with her in the house as she would suddenly become very attention seeking and I would get very annoyed at this. Even if I set her up with something or put on a DVD, she would want me to sit with her and tantrum when I did not and would do that everyday in spite of being told no. She would also follow me and get under my feet. Even clearing the kitchen or cooking was an ordeal. DD was easier with DH at that age and would have complied more easily with him, which is normal.
It is now the school holidays and I spent an hour trying to get stuff done. I decided to dust their rooms. In the middle, DD started her homework which she could not do so I got constant questions and she got frustrated. I had told her to do it later when I was not busy but she didn't listen. I had to stop to prevent her from bashing the computer in temper. DS wanted his bath then wanted me to play with him. Despite repeated rebuffals he kept asking. Meanwhile the washing machine was bleeping because that had finished and the phone rang in the middle of it, too. Then DS wanted out of the bath and help drying himself.
They are quiet now but I still haven't showered DD but sometimes when they have stopped demanding/fighting/screaming, you need a breather. I definitely saw pre-school time as that back in the day. It is literally so full on when you are doing it, you really do need that time. And housework is such a physical task. On top of looking after a young child, it is really tiring and you certainly wouldn't want do to do it all day. It is easy to lose motivation, too.
Nevertheless, I hear you OP, it sounds as if he needs to take on a bit more and really tightly structure his day. You can't not. Otherwise, time can really run away with you.