I think there are a few separate issues here that are getting conflated.
Would you be unreasonable asking him to take a sick day - yes, because he's not sick but you've acknowledged that and said you used the wrong word.
Would you be unreasonable asking him to take unpaid leave, dependent's leave, emergency leave, holiday day etc. - it depends on the extent of your injuries and the nature of his work.
My OH broke his arm and was stuck with the car, so I had to rearrange all my meetings to go get him and drive the car back.
Fortunately on that day I was able to do so. That said, there have been other days where I just wouldn't have been able to.
I've had meetings that can't be moved, big client facing days, business travel trips and even the odd employer who would get shirty (rare).
On one of those days it would have been a right royal pain, but we'd have called in on friends / family to help.
So... maybe YABU, maybe he is dodging responsibilities.
To me it sounds like your injuries are very painful, but there are ways you can get around them e.g. taxi for school run etc, so he may be better off saving up employer goodwill for genuine emergencies.
But on other days where it's lots of conf calls etc, I might easily be able to WFH and help out. And I would do this even if injuries are minor to be nice.
So as I said. Depends 1) on how badly incapacitated you are and 2) the nature of his work and how busy his work day is.
Finally - was he unreasonable to sit on his arse watching telly while you struggled to cook, clean and sort the kids. Yes. He was absolutely outrageous and I would have been fuming.
Firstly he should have bloody offered. Then, when he didn't you should have insisted (annoying to have to manage him but nonetheless).