My mom broke her arm last Wednesday and has it in a cast and sling. Is it unreasonable that I don't want her doing housework?

Very reasonable but Mom does not see it that way, She does not want to feel disabled and dependant.
Carrying the rubbish down the stairs was not very smart but she is not going to stop.
What I would suggest is that you give her a list of things that will be helpful while you are at work but not hazardous to her.
She can dust, put laundry in the washer as long as she does not need to go down the basement steps, Bring in the mail and newspaper, pull some weeds, those sort of things she is capable of.
Does she have a medical alert button. Could you ask a neighbor to look in on her or maybe invite her for coffee

She'll show you, right? Does she "get" that a broken hip could land her in a nursing home? Has she always been stubborn or is this perhaps symptomatic of cognitive decline.

It can be both, of course. Perhaps if her doctor told her she has to rest?

I had a similar situation with my mom. Bottom line was she was going to do what she was going to do, partly because she needed to be needed, partly because no kid was going to tell her what to do. My wise husband told me I needed to let it go and quit fussing at her and stressing myself out. So I told her I was not going to say another word and I didn't. I felt much calmer not trying to control the uncontrollable.

I know you don't want her to get hurt but sometimes we need to just let it go.

Bronnie, I think it is good that Mom is trying to help, just as long as she is not feeling pain in her arm. Got to admit, carrying the rubbish down the stairs was another accident waiting to happen.... thankfully she made it down safely and probably thought after the fact, it's wasn't a bright idea.

After having an arm in a sling for two or three weeks that arm becomes terrible stiff, to a point where it is hard to use.... chances are Mom will need some type of physical therapy to get the arm back into a full range of motion.

That happened to me when I fell and broke my upper arm. But I didn't have a cast, just a sling. Guess it depends on where the break happens. It hard enough if the break is on your primary arm, trying to write, eat, comb your hair, attempting to put on clothes, bras were next to impossible !!

Bronnie, what can she do sitting down? Folding laundry has been suggested by others for parents, sometimes in dementia situations, but it can also be done when range of motion is limited.

Can she set the table? Sort out the junk mail? I can't think of other seated activities now.