I have to say that we do have this issue a bit in my house too.
Partner runs a company with another guy and unless one of them is there it can't run, staff can't be let in etc, although the main issue for my partner is if he isn't there they don't make money. Anyway this has cause problems in past with us. As much as I understand this when you're ill it does go out the window a bit as you feel so vulnerable.
I was in hospital recently and typically the other guy who runs the business was on holiday so my OH had to open up, sort the kids out by dropping them off at breakfast club and then going to work. I didn't mind having him not there whilst I was in as I didn't feel that ill and my friend was coming in plus he fidgets so much and taps the Hosp bed with his foot it gives me utter rage
He just collected the kids and came on in with them, it worked ok. If I had been in your position OP he probably would have still gone to work but made sure I was ok, sorted children out or just taken day off if I was really bad.
He doesn't have time off ever, he's even gone to work with a plastered wrist and plastered leg (not at the same time!).
We did have a issue once in the past similair to the OP and after that I made it perfectly clear if I am ill, and feel that bad I don't want to hear about work, family comes first and he can deal with work later.