If it were my house, I'd:

1. Remove each tread and cut it to be flush with the riser beneath on both sides.

2. Cover up both sides where the steps meet the drywall with a 1x8 or 1x10 scribed to each step (think baseboard).

It even looks like you have a gap wide enough on the right side that if you pull all the treads off, you can slide the 1x8 or 1x10 between the wall and steps so that you wouldn't have to scribe it. Then replace the treads cut to the proper width. You'll likely still have to scribe the left side, though, which will take some skill to get right.

^^ Pretty good advice, I like that plan.

The treads are already oak it looks like, so why cover with more oak when you could tidy it up and use it as is.

Question, on the right side where the drywall tapers, is that wall plumb? If not, I could see a reason to fix that.

Is that piece of trim/railing really needed? Or was it just to cover a sloppy drywall joint? If the latter, you can remove it and do the joint better. The other possibility is that it's two different thicknesses of drywall meeting up, and the trim was supposed to cover that up. I'd lose that trim if I could.

The drywall at the top of the staircase is set on the steps. The bottom section of drywall is set just outside the steps. The drywall sections don’t match up vertically, and the piece of trim hides the gaping joint.

Photos

Is the backside (framing) of the stairs accessible? If so, post of pic of the framing. I have an idea....

i wouldnt replace the treads, I would be sanding and stripping and then restaining etc them. That is some very nice looking wood on them now.

This is underneath the left side of the staircase if you were descending. There are two 2x4s that are perpendicular to each other instead of in a straight line to form a straight wall.

As per shead's second suggestion, I would consider narrowing the treads on the right side, but without removing them. Then insert what would appear as a closed stringer to the right.

For the left side, I'd add trim to all the steps.

Be sure to add a handrail when you're done!

This is the right side. There is a supporting post and lots of hvac stuff.

Well, bummer. I was hoping your left stringer was indented enough that you could trim the risers and treads on the left enough to slide a closed stringer board (Thanks, worthy, for the correct terminology!) between the studs and stairs.

In any case, I'd follow worthy's advice and put a closed stronger board on the right and either trim to cover the gaps on the left or retrofit and closed stringer board that would actually rest on top of the treads and risers versus being inserted between the stair and studs like the right side.

"My guess is that the contractor who put them up did not want to make all those cuts to fit it nicely on the stairs."

I think what you are dealing with is a remodel or two done by a hack contractor or the former homeowner, and whoever did it either wanted to do it the easiest way or was somewhat incompetent, or both. I think the existing stairs were butchered in the remodel process, not that they were installed that way.

I know that doesn't help you fix this, but it sounded like you were puzzled that it was done like this so I thought I'd opine.

Yep, probably a mix of lazy and lacking in skills. But he did pull permits for the electric. I think.

When we remodeled they had to build the underneath with fire blocking in place. The whole space had to be drywalled and blocking in the cavities of the framing. Just make sure it’s all fire safe!