Removing the scrolls should be easy.

Removing the rounded side of the stair will shorten the overhang of the tread and that may look out of proportion. You will also have to cut off the mitered angle at the front.

I would start step-wise, no pun intended, and remove the scrolls first, then reevaluate.

Difficult to see the stair from here.

Thanks palimpsest. Yes, that was my plan to pull off a scroll on one of the top steps and then think about where to go from there. If I pull of the rounded piece on the outside edge, I could always replace it with a straight piece and I think that would make it more contemporary looking.

I'm also wondering what to do with the skirting on the outside edge (under the scrolls) and also on the inside (against the wall). Its fir (or pine?), not oak like the stairs and I think it looks bad. Wondering if it would look funny is it was painted, given that all of the other woodwork in the house is stained.

Be aware that if you pull off the scroll pieces they're likely trimming over a joint that you don't want exposed. So you may need to replace them with something, albeit in a different style. It also looks like the scroll pieces go underneath the cove molding that's under the treads. That will make for more difficult removal, and it may take the cove molding off with it.

Painted risers/trim with stained treads is a perfectly acceptable modern look. Here's what we did with ours:

Squaring off the treads would be tough to do in place. You could remove them and square them pretty easily with a router, but the remove/reinstall would be a pain. I think the curved edges look perfectly fine.

Consider replacing the square balusters with wrought iron ones. As long as the existing balusters are no more than 4" OC, you could use the existing holes and just swap them out.