I see some of my DH in your description - I think it's a 'man' problem in general, as much as it is for your DH - some men just don't 'see' the things that need doing.
DH & I both have our own businesses and work from home In theory we both work about 4 days over 5 days and evenings. In theory we 'share' the after school childcare and school runs.
In practice, when I'm doing 'after school' childcare I feed them tea, oversee homework, pack bags for next morning, do a couple of loads of laundry & folding/ putting away, listen to reading, hear spellings and cook our dinner.
When DH is doing after school childcare he brings the kids home from school, gives them some squash and a biscuit, and then buggers off upstairs to his office until about 6 pm.
It is a constant source of argument between us, and like your DH, he won't hear any suggestions about what else he should be doing! As a result I am spending about 2 days a week working, and end up doing everything else!
The only partial success I had was by putting together an 'after school checklist' which listed all the stuff he HAD to do with the kids on a particular day, and then he reluctantly followed it (for a couple of weeks at least )